Hardness of Approximating the Minimum Distance of a Linear Code

Transcription

1 Hadness of Appoximating the Minimum Distance of a Linea Code Iya Dume Daniee Micciancio Madhu Sudan Abstact We show that the minimum distance d of a inea code is not appoximabe to within any constant facto in andom poynomia time (RP), uness NP (nondeteministic poynomia time) equas RP. We aso show that the minimum distance is not appoximabe to within an additive eo that is inea in the bock ength n of the code. Unde the stonge assumption that NP is not contained in RQP (andom quasi-poynomia time), we show that the minimum distance is not appoximabe to within the facto 2 og1 ɛ (n), fo any ɛ > 0. Ou esuts hod fo codes ove any finite fied, incuding binay codes. In the pocess we show that it is had to find appoximatey neaest codewods even if the numbe of eos exceeds the unique decoding adius d/2 by ony an abitaiy sma faction ɛd. We aso pove the hadness of the neaest codewod pobem fo asymptoticay good codes, povided the numbe of eos exceeds (2/3)d. Ou esuts fo the minimum distance pobem stengthen (though using stonge assumptions) a pevious esut of Vady who showed that the minimum distance cannot be computed exacty in deteministic poynomia time (P), uness P NP. Ou esuts ae obtained by adapting poofs of anaogous esuts fo intege attices due to Ajtai and Micciancio. A citica component in the adaptation is ou use of inea codes that pefom bette than andom (inea) codes. Index Tems Computationa compexity, NP-hadness, inea codes, dense codes, appoximation agoithms, minimum distance pobem, eativey nea codewod pobem, bounded distance decoding. i.e., A is a inea subspace of F n q ove base fied F q. Fo such a code, the infomation content (i.e., the numbe k og q A of infomation symbos that can be encoded with a codewod 1 ) is just its dimension as a vecto space and the code can be compacty epesented by a k n geneato matix A F k n q of ank k such that A {xa x F k q }. An impotant popety of a code is its minimum distance. Fo any vectos x, y Σ n, the Hamming weight of x is the numbe wt(x) of nonzeo coodinates of x. The weight function wt( ) is a nom, and the induced metic d(x, y) wt(x y) is caed the Hamming distance. The (minimum) distance d(a) of the code A is the minimum Hamming distance d(x, y) taken ove a pais of distinct codewods x, y A. Fo inea codes it is easy to see that the minimum distance d(a) equas the weight wt(x) of the ightest nonzeo codewod x A \ {0}. If A is a inea code ove F q with bock ength n, ank k and minimum distance d, then it is customay to say that A is a inea [n, k, d] q code. Thoughout the pape we use the foowing notationa conventions: matices ae denoted by bodface uppecase Roman ettes (e.g., A,C), the associated inea codes ae denoted by the coesponding caigaphic ettes (e.g., A, C), and we wite A[n, k, d] q to mean that A is a inea code ove F q with bock ength n, infomation content k and minimum distance d. I. INTRODUCTION In this pape we study the computationa compexity of two centa pobems fom coding theoy: (1) The compexity of appoximating the minimum distance of a inea code and (2) The compexity of eo-coection in codes of eativey age minimum distance. An eo-coecting code A of bock ength n ove a q-ay aphabet Σ is a coection of stings (vectos) fom Σ n, caed codewods. Fo a codes consideed in this pape, the aphabet size q is aways a pime powe and the aphabet Σ F q is the finite fied with q eement. A code A F n q is inea if it is cosed unde addition and mutipication by a scaa, Peiminay vesions of this wok wee pesented at the 40th IEEE Annua Symposium on Foundations of Compute Science (FOCS), New Yok, USA, Octobe 1999, and the IEEE Intenationa Symposium on Infomation Theoy (ISIT), Soento, Itay, June I. Dume is with the Coege of Engineeing, Univesity of Caifonia at Riveside. Riveside, CA 92521, USA (emai: dume@ee.uc.edu). Reseach suppoted in pat by NSF gant NCR D. Micciancio is with the Depatment of Compute Science and Engineeing, Univesity of Caifonia, San Diego, 9500 Giman Dive, Mai Code 0114, La Joa, CA 92093, USA (emai: daniee@cs.ucsd.edu). Reseach suppoted in pat by NSF Caee Awad CCR M. Sudan is with the Depatment of Eectica Engineeing and Compute Science, Massachusetts Institute of Technoogy. 545 Technoogy Squae, Cambidge, MA 02139, USA (emai: madhu@mit.edu). Reseach suppoted in pat by a Soan Foundation Feowship, an MIT-NEC Reseach Initiation Gant and NSF Caee Awad CCR A. The Minimum Distance Pobem Thee of the fou centa paametes associated with a inea code, namey n, k and q, ae evident fom its matix epesentation. The minimum distance pobem (MINDIST) is that of evauating the fouth namey given a geneato matix A F k n q find the minimum distance d of the coesponding code A. The minimum distance of a code is obviousy eated to its eo coection capabiity (d 1)/2 and theefoe finding d is a fundamenta computationa pobem in coding theoy. The pobem gains even moe significance in ight of the fact that ong q-ay codes chosen at andom give the best paametes known fo any q < 46 (in paticua, fo q 2) 2. A poynomia time agoithm to compute the distance woud be the idea soution to the pobem, as it coud be used to constuct good eo coecting codes by choosing a geneato matix at andom and checking if the associated code has a age minimum distance. Unfotunatey, no such agoithm is known. The compexity of this pobem (can it be soved in poynomia time o not?) 1 Thoughout the pape, we wite og fo the ogaithm to the base 2, and og q when the base q is any numbe possiby diffeent fom 2. 2 Fo squae pime powes q 49, inea AG codes can pefom bette than andom ones [21] and ae constuctibe in poynomia time. Fo a othe q 46 it is sti possibe to do bette than andom codes, howeve the best known pocedues to constuct them un in exponentia time [24].

2 2 was fist expicity questioned by Beekamp, McEiece and van Tibog [5] in 1978 who conjectued it to be NP-compete. This conjectue was finay esoved in the affimative by Vady [23], [22] in 1997, poving that the minimum distance cannot be computed in poynomia time uness P NP. ([23], [22] aso give futhe motivations and detaied account of pio wok on this pobem.) To advance the seach of good codes, one can eax the equiement of computing d exacty in two ways: Instead of equiing the exact vaue of d, one can aow fo appoximate soutions, i.e., an estimate d that is guaanteed to be at east as big, but not much bigge than the tue minimum distance d. (E.g., d d γd fo some appoximation facto γ). Instead of insisting on deteministic soutions that aways poduce coect (appoximate) answes, one can conside andomized agoithms such that d γd ony hods in a pobabiistic sense. (Say d γd with pobabiity at east 1/2.) 3 Such agoithms can sti be used to andomy geneate eativey good codes as foows. Say we want a code with minimum distance d. We pick a geneato matix at andom such that the code is expected to have minimum distance γd. Then, we un the pobabiistic distance appoximation agoithm many times using independent coin tosses. If a estimates etuned by the distance appoximation agoithm ae at east γd, then the minimum distance of the code is at east d with vey high pobabiity. In this pape, we study these moe eaxed vesions of the minimum distance pobem and show that the minimum distance is had to appoximate (even in a pobabiistic sense) to within any constant facto, uness NP RP (i.e., evey pobem in NP has a poynomia time pobabiistic agoithm that aways ejects NO instances and accepts YES instances with high pobabiity). Unde the stonge assumption that NP does not have andom quasi-poynomia 4 time agoithms (RQP), we pove that the minimum distance of a code of bock ength n is not appoximabe to within a facto of 2 og(1 ɛ) n fo any constant ɛ > 0. (This is a natuay occuing facto in the study of the appoximabiity of optimization pobems see the suvey of Aoa and Lund [4].) Ou methods adapt the poof of the inappoximabiity of the shotest attice vecto pobem (SVP) due to Micciancio [18] (see aso [19]) which in tun is based on Ajtai s poof of the hadness of soving SVP [2]. B. The Eo Coection Pobem In the pocess of obtaining the inappoximabiity esut fo the minimum distance pobem, we aso shed ight on the genea eo-coection pobem fo inea codes. The simpest fomuation is the Neaest Codewod Pobem (NCP) (aso known as the maximum ikeihood decoding pobem ). Hee, the input instance consists of a geneato matix A F k n q and a eceived wod x F n q and the goa is to find the neaest 3 One can aso conside andomized agoithms with 2-sided eo, whee aso the owe bound d d hods ony in a pobabiistic sense. A ou esuts can be easiy adapted to agoithms with 2-sided eo. 4 f(n) is quasi-poynomia in n if it gows sowe than 2 ogc n fo some constant c. codewod y A to x. A moe eaxed vesion is to estimate the minimum eo weight d(x, A) that is the distance d(x, y) to the neaest codewod, without necessaiy finding codewod y. The NCP is a we-studied pobem: Beekamp, McEiece and van Tibog [5] showed that it is NP-had (even in its weight estimation vesion); and moe ecenty Aoa, Babai, Sten and Sweedyk [3] showed that the eo weight is had to appoximate to within any constant facto uness P NP, and within facto 2 og(1 ɛ) n fo any ɛ > 0, uness NP QP (deteministic quasi-poynomia time). This atte esut has been ecenty impoved to inappoximabiity within 2 O(og n/ og og n) n 1/O(og og n) unde the assumption that P NP by Dinu, Kinde, Raz and Safa [9], [8]). On the positive side, NCP can be tiviay appoximated within a facto n. Genea, non-tivia appoximation agoithms have been ecenty discoveed by Beman and Kapinski [6], who showed that (fo any finite fied F q ) NCP can be appoximated within a facto ɛn/ og n fo any fixed ɛ > 0 in pobabiistic poynomia time, and ɛn in deteministic poynomia time. Howeve the NCP ony povides a fist cut at undestanding the eo-coection pobem. It shows that the eo-coection pobem is had, if we ty to decode evey inea code egadess of the eo weight. In contast, the positive esuts fom coding theoy show how to pefom eo-coection in specific inea codes coupted by eos of sma weight (eative to the code distance). Thus the hadness of the NCP may come fom one of two factos: (1) The pobem attempts to decode evey inea code and (2) The pobem attempts to ecove fom too many eos. Both issues have been aised in the iteatue [23], [22], but ony the fome has seen some pogess [7], [10], [20]. One pobem that has been defined to study the atte phenomenon is the Bounded distance decoding pobem (BDD, see [23], [22]). This is a specia case of the NCP whee the eo weight is guaanteed (o pomised ) to be ess than d(a)/2. This case is motivated by the fact that within such a distance, thee may be at most one codewod and hence decoding is ceay unambiguous. Aso this is the case whee many of the cassica eo-coection agoithms (fo say BCH codes, RS codes, AG codes, etc.) wok in poynomia time. C. Reativey Nea Codewod Pobem To compae the genea NCP, and the moe specific BDD pobem, we intoduce a paameteized famiy of pobems that we ca the Reativey Nea Codewod Pobem (RNC). Fo ea ρ, RNC (ρ) is the foowing pobem: Given a geneato matix A F k n q of a inea code A of (not necessaiy known) minimum distance d, an intege t with the pomise that t < ρ d, and a eceived wod x F n q, find a codewod within distance t fom x. (The agoithm may fai if the pomise is vioated, o if no such codewod exists. In othe wods, in contast to the papes [3] and [5], the agoithm is expected to wok ony when the eo weight is imited in popotion to the code distance.) Both the neaest codewod pobem (NCP) and the bounded distance decoding pobem (BDD) ae specia cases of RNC (ρ) : NCP RNC ( ) whie BDD RNC ( 1 2 ). Ti

3 3 ecenty, not much was known about RNC (ρ) fo constants ρ <, eave aone ρ 1 2 (i.e., the BDD pobem). No finite uppe bound on ρ can be easiy deived fom Aoa et a. s NP-hadness poof fo NCP [3]. (In othe wods, thei poof does not seem to hod fo RNC (ρ) fo any ρ <.) It tuns out, as obseved by Jain et a. [14], that Vady s poof of the NP-hadness of the minimum distance pobem aso shows the NP-hadness of RNC (ρ) fo ρ 1 (and actuay extends to some ρ 1 o(1)). In this pape we significanty impove upon this situation, by showing NP-hadness (unde andomized eductions) of RNC (ρ) fo evey ρ > 1 2 binging us much cose to an eventua (negative?) esoution of the bounded distance decoding pobem. D. Oganization The est of the pape is oganized as foows. In Section II we pecisey define the coding pobems studied in this pape, intoduce some notation, and biefy oveview the andom eduction techniques and coding theoy notions used in the est of the pape. As expained in Section II ou poofs ey on coding theoetic constuctions that might be of independent inteest, namey the constuction of codes containing unusuay dense custes. These constuctions constitute the main technica contibution of the pape, and ae pesented in Section III (fo abitay inea codes) and Section VI (fo asymptoticay good codes). The hadness of the eativey nea codewod pobem and the minimum distance pobem ae poved in Sections IV and V espectivey. Simia hadness esuts fo asymptoticay good codes ae pesented in Section VII. The eade mosty inteested in computationa compexity may want to skip Section III at fist eading, and jump diecty to the NP-hadness esuts in Sections IV and V. Section VIII concudes with a discussion of the consequences and imitations of the poofs given in this pape, and eated open pobems. A. Appoximation Pobems II. BACKGROUND In ode to study the computationa compexity of coding pobems, we fomuate them in tems of pomise pobems. A pomise pobem is a geneaization of the famiia notion of decision pobem. The diffeence is that in a pomise pobem not evey sting is equied to be eithe a YES o a NO instance. Given a sting with the pomise that it is eithe a YES o NO instance, one has to decide which of the two sets it beongs to. The foowing pomise pobem captues the hadness of appoximating the minimum distance pobem within a facto γ. Definition 1 Minimum Distance Pobem: Fo pime powe q and appoximation facto γ 1, an instance of GAPDIST γ,q is a pai (A, d), whee A F k n q and d Z +, such that (A, d) is a YES instance if d(a) d. (A, d) is a NO instance if d(a) > γ d. In othe wods, given a code A and an intege d with the pomise that eithe d(a) d o d(a) > γ d, one must decide which of the two cases hods tue. The eation between appoximating the minimum distance of A and the above pomise pobem is easiy expained. On the one hand, if one can compute a γ-appoximation d [d(a), γ d(a)] to the minimum distance of the code, then one can easiy sove the pomise pobem above by checking whethe d γ d o d > γ d. On the othe hand, assume one has a decision oace O that soves the pomise pobem above. Then, the minimum distance of a given code A can be easiy appoximated using the oace as foows. Notice that O(A, n) aways etuns YES whie O(A, 0) aways etuns NO. Using binay seach, one can efficienty find a numbe d such that O(A, d) YES and O(A, d 1) NO. 5 This means that (A, d) is not a NO instance and (A, d 1) is not a YES instance, and the minimum distance d(a) must ie in the inteva [d, γ d]. Simiay we can define the foowing pomise pobem to captue the hadness of appoximating RNC (ρ) within a facto γ. Definition 2 Reativey Nea Codewod Pobem: Fo pime powe q, and factos ρ > 0 and γ 1, an instance of GAPRNC γ,q (ρ) is a tipe (A, v, t), whee A Fq k n, v F n q and t Z +, such that t < ρ d(a) and 6 (A, v, t) is a YES instance if d(v, A) t. (A, v, t) is a NO instance if d(v, A) > γt. It is immediate that the pobem RNC (ρ) gets hade as ρ inceases, since changing ρ has the ony effect of weakening the pomise t < ρ d(a). RNC (ρ) is the hadest when ρ in which case the pomise t < ρd(a) is vacuousy tue and we obtain the famiia (pomise vesion of) the neaest codewod pobem: Definition 3 Neaest Codewod Pobem: Fo pime powe q and γ 1, an instance of GAPNCP γ,q is a tipe (A, v, t),, v F n q and t Z +, such that (A, v, t) is a YES instance if d(v, A) t. (A, v, t) is a NO instance if d(v, A) > γ t. The pomise pobem GAPNCP γ,q is NP-had fo evey constant γ 1 (cf. [3] 7 ), and this esut is citica to ou hadness esut(s). We define one ast pomise pobem to study the hadness of appoximating the minimum distance of a code with inea additive eo. Definition 4: Fo τ > 0 and pime powe q, et A F k n q GAPDISTADD τ,q be the pomise pobem with instances (A, d), whee A F k n q and d Z +, such that (A, d) is a YES instance if d(a) d (A, d) is a NO instance if d(a) > d + τ n. B. Random Reductions and Techniques The main esut of this pape (see Theoem 22) is that appoximation pobem GAPDIST γ,q is NP-had fo any constant facto γ 1 unde poynomia evese unfaithfu andom eductions (RUR-eductions, [15]), and fo γ 2 og(1 ɛ) n 5 By definition, the oace can give any answe if the input is neithe a YES instance no a NO one. So, it woud be wong to concude that (A, d 1) is a NO instance and (A, d) is a YES one. 6 Sticty speaking, the condition t < ρ d(c A ) is a pomise and hence shoud be added as a condition in both the YES and NO instances of the pobem. 7 To be pecise, Aoa et a. [3] pesent the esut ony fo binay codes. Howeve, thei poof is vaid fo any aphabet. An atenate way to obtain the esut fo any pime powe is to use a ecent esut of Håstad [13] who states his esut in inea ageba (athe than coding-theoetic) tems. We wi state and use some of the additiona featues of the atte esut in Section VII.

4 4 n 1/ ogɛ (n) it is had unde quasi-poynomia RUR-eductions. These ae pobabiistic eductions that map NO instances aways to NO instances and YES instances to YES instances with high pobabiity. In paticua, given a secuity paamete s, a eductions pesented in this pape waant that YES instances ae popey mapped with pobabiity 1 q s in poy(s) time. 8 The existence of a (andom) poynomia time agoithm to sove the had pobem woud impy NP RP (andom poynomia time), i.e., evey pobem in NP woud have a pobabiistic poynomia time agoithm that aways ejects NO instances and accepts YES instances with high pobabiity. 9 Simiay, hadness fo NP unde quasi-poynomia RUR-eductions impies that the had pobem cannot be soved in RQP uness NP RQP (andom quasi-poynomia time). Theefoe, simiay to a pope NP-hadness esut (obtained unde deteministic poynomia eductions), hadness unde poynomia RUReductions aso gives evidence of the intactabiity of a pobem. In ode to pove these esuts, we fist study the pobem GAPRNC (ρ) γ,q. We show that the eo weight is had to appoximate to within any constant facto γ and fo any ρ > 1/2 uness NP RP (see Theoem 16). By using γ 1/ρ, we immediatey educe ou eo-coection pobem GAPRNC (ρ) γ,q to the minimum distance pobem GAPDIST γ,q fo any constant γ < 2. We then use poduct constuctions to ampify the constant and pove the caimed hadness esuts fo the minimum distance pobem. The hadness of GAPRNC (ρ) γ,q fo ρ > 1/2 is obtained by adapting a technique of Micciancio [18], which is in tun based on the wok of Ajtai [2] (hencefoth Ajtai-Micciancio). They conside the anaogous pobem ove the integes (athe than finite fieds) with Hamming distance epaced by Eucidean distance. Much of the adaptation is staightfowad; in fact, some of the poofs ae even easie in ou case due to the use of finite fieds. The main hude tuns out to be in adapting the foowing combinatoia pobem consideed and soved by Ajtai- Micciancio: Given an intege k constuct, in poy(k) time, integes,, an -dimensiona attice L (i.e., a subset of Z cosed unde addition and mutipication by an intege) with minimum (Eucidean) distance d > /ρ and a vecto v Z such that the (Eucidean) ba of adius aound v contains at east 2 k vectos fom L (whee ρ < 1 is a constant independent of k). In ou case we ae faced with a simia pobem with Z epaced by F q and Eucidean distance epaced by Hamming distance. The Ajtai-Micciancio soution to the above pobem in- 8 Hee, and in the est of the pape, we use notation poy(n) to denote any poynomiay bounded function of n, i.e., any function f(n) such that f(n) O(n c ) fo some constant c indepentent of n. 9 Notice that we ae unabe to pove that the existence of a poynomia time appoximation agoithm impies the stonge containment NP ZPP (ZPP is the cass of decision pobems L such that both L and its compement ae in RP, i.e., ZPP RP corp), as done fo exampe in [12]. The diffeence is that [12] poves hadness unde unfaithfu andom eductions (UR-eductions [15], i.e. eductions that ae aways coect on YES instances and often coect on NO instances). Hadness unde UR-eductions impies that no poynomia time agoithm exists uness NP corp, and theefoe RP NP corp. It immediatey foows that corp RP and NP RP corp ZPP. Howeve, in ou case whee RUR-eductions ae used, we can ony concude that RP NP RP, and theefoe NP RP, but it is not cea how to estabish any eation invoving the compementay cass corp and ZPP. voves numbe-theoetic methods and does not tansate to ou setting. Instead we show that if we conside a inea code whose pefomance (i.e., tade-off between ate and distance) is bette than that of a andom code, and pick a andom ight vecto in F n q, then the esuting constuction has the equied popeties. We fist sove this pobem ove sufficienty age aphabets using high ate Reed-Soomon (RS) codes. (See next subsection fo the definition RS codes. This same constuction has been used in the coding theoy iteatue to demonstate imitations to the ist-decodabiity of RS codes [16].) We then tansate the esut to sma aphabets using the we-known method of concatenating codes [11]. In the second pat of the pape, we extend ou methods to addess asymptoticay-good codes. We show that even fo such codes, the Reativey Nea Codewod pobem is had uness NP equas RP (see Theoem 31), though fo these codes we ae ony abe to pove the hadness of GAPRNC (ρ) γ,q fo ρ > 2/3. Finay, we tansate this to a esut (see Theoem 32) showing that the minimum distance of a code is had to appoximate to within an additive eo that is inea in the bock ength of the code. C. Coding Theoy Fo a though intoduction to coding theoy the eade is efeed to [17]. Hee we biefy eview some basic esuts as used in the est of the pape. Reades with an adequate backgound in coding theoy can safey skip to the next section. A cassica pobem in coding theoy is to detemine fo given aphabet F q, bock ength n and dimension k, what is the highest possibe minimum distance d such that thee exists a [, m, d] q inea code. The foowing two theoems give we known uppe and owe bounds fo d. Theoem 5 Gibet-Vashamov bound: Fo finite fied F q, bock ength and dimension m, thee exists a inea code A[, m, d] q with minimum distance d satisfying q m B(0, d 1) 1 (1) whee B(0, d 1) d 1 ( ) 0 (q 1) is the voume of the dimensiona Hamming sphee of adius d 1. Fo any m, the smaest d such that (1) hods tue is denoted d m. (Notice that the definition of d m depends both on the infomation content m and the bock ength. Fo bevity, we omit fom the notation, as the bock ength is usuay cea fom the context.) See [17] fo a poof. The uppe bound on d given by the Gibet-Vashamov theoem (GV bound, fo shot) is not effective, i.e., even if the theoem guaantees the existence of inea codes with a cetain minimum distance, the poof of the theoem empoys a geedy agoithm that uns in exponentia time, and we do not know any efficient way to find such codes fo sma aphabets q < 49. Inteestingy, andom inea codes meet the GV bound, i.e., if the geneating matix of the code is chosen unifomy at andom, then the minimum distance of the esuting code satisfies the GV bound with high pobabiity. Howeve, given a andomy chosen geneato matix, it is not cea how to efficienty check whethe the coesponding code meets the GV bound o not. We now give a owe bound on d.

5 5 Theoem 6 Singeton bound: Fo evey inea code C[, m, d] q, the minimum distance d is at most d m + 1. See [17] fo a poof. A code C[, m, d] q is caed Maximum Distance Sepaabe (MDS) if the Singeton bound is satisfied with equaity, i.e., if d m + 1. Reed-Soomon codes ae an impotant exampe of MDS codes. Fo any finite fied F q, and dimension m q 1, the Reed-Soomon code (RS code) G[q 1, m, q m] q can be defined as the set of (q 1)-dimensiona vectos obtained evauating a q m poynomias p(x) c 0 + c 1 x + c m 1 x m 1 F q [x] of degee ess than m at a nonzeo points x F q \ {0}. Extended Reed-Soomon codes G[q, m, q m + 1] q ae defined simiay, but evauating the poynomias at a points x F q, incuding 0. Since a nonzeo poynomia p of degee (m 1) can have at most (m 1) zeos, then evey nonzeo (extended) RS codewod has at most m 1 zeo positions, so it s Hamming weight is at east q m (esp. q m + 1). This poves that (extended) Reed-Soomon codes ae maximum distance sepaabe. Extended RS codes can be futhe extended consideing the homogeneous poynomias p(x, y) m 1 i1 a ix i y m 1 i, and evauating them at a points of the pojective ine {(x, 1): x F q } {(1, 0)}. This inceases both the bock ength and the minimum distance by one, giving twice extended RS codes G[q + 1, m, q m + 2] q. Anothe famiy of codes we ae going to use ae the Hadamad codes H[q c 1, c, q c q c 1 ] q. In these codes, each codewod coesponds to a vecto x F c q. The codewod associated to x is obtained by evauating a nonzeo inea functions φ : F c q F q at x. Since thee ae q c inea functions in c vaiabes (incuding the zeo function), the code has bock ength q c 1. Notice that any nonzeo vecto x F c q is mapped to 0 by exacty a 1/q faction of the inea functions φ : F c q F q (incuding the identicay zeo function). So the minimum distance of the code is q c q c 1. In Section VI we wi aso use Agebaic Geometic codes (AG codes). The definition of these codes is beyond the scope of the pesent pape, and the inteested eade is efeed to [21]. An impotant opeation used to combine codes is the concatenating code constuction of [11]. Let A and B be two inea codes ove aphabets F q k and F q, with geneating matices A F m and B F n k q k q. A is caed the oute code, and B is caed the inne code. Notice that the dimension of the inne code equas the dimension of the aphabet of the oute code F q k, viewed as a vecto space ove base fied F q. The idea is to epace evey component x i of oute codewod [x 1,..., x ] A F q with a coesponding inne codewod k φ(x i ) B. Moe pecisey, et α 1,..., α k F q k be a basis fo F q k as a vecto space ove F q and et φ : F q k B be the (unique) inea function such that φ(α i ) b i fo a i 1,..., k. The concatenation function B: F q F n k q is given by [x 1,..., x n ] B [φ(x 1 ),..., φ(x n )]. Function B is extended to sets of vectos in the usua way X B {x B: x X}, and concatenated code A B is just the esut of appying the concatenation function B to set A. It is easy to see that if A is a [, m, d] q k code and B is a [n, k, t] q code, then the concatenation A B is a [n, km, dt] q code with geneato matix C Fq n km given by c i+jk (α i a j+1 ) B. fo a i 1,..., k and j 0,..., m 1. III. DENSE CODES In this section we pesent some genea esuts about codes with a specia density popety (to be defined), and thei agoithmic constuction. The section cuminates with the poof of Lemma 15 which shows how to efficienty constuct a gadget that wi be used in Section IV in ou NP-hadness poofs. Lemma 15 is in fact the ony esut fom this section diecty used in the est of the pape (with the exception of Section VI which extends the esuts of this section to asymptoticay good codes). The eade mosty inteested in computationa compexity issues, may want to skip this section at fist eading, and efe to Lemma 15 when used in the poofs. A. Genea oveview Let B(v, ) {x F q d(v, x) } be the ba of adius centeed in v F q. In this section, we wish to find codes C[, m, d] q that incude mutipe codewods in some ba(s) B(v, ) of eativey sma adius ρd, whee ρ is some positive ea numbe. Obviousy, the pobem is meaningfu ony fo ρ 1/2, as any ba of adius < d/2 cannot contain moe than a singe codewod. Beow we pove that fo any ρ > 1/2 it is actuay possibe to buid such a code. These codes ae used in the seque to pove the hadness of GAPRNC (ρ) γ,q by eduction fom the neaest codewod pobem. Of paticua inteest is the case < d (i.e., ρ < 1), as the coesponding hadness esut fo GAPRNC can be tansated into an inappoximabiity esut fo the minimum distance pobem. We say that a code C[, m, d] q is (ρ, k)-dense aound v F q if the ba B(v, ρd ) contains at east q k codewods. We say that C is (ρ, k)-dense if it is (ρ, k)-dense aound v fo some v. We want to detemine fo what vaues of the paametes ρ, k,, m, d thee exist (ρ, k)-dense codes. The technique we use is pobabiistic: we show that thee exist [, m, d] q codes such that the expected numbe of codewods in a andomy chosen sphee B(v, ρd) is at east q k. It easiy foows, by a simpe aveaging agument, that thee exists a cente v such that the code is (ρ, k)-dense aound v. Let µ C () Exp [ C B(v, ) ] v F q x C P {x B(v, )} v F q be the expected numbe of codewods in B(v, ) when the cente v is chosen unifomy at andom fom F q. Notice that µ C () P {v B(x, )} v F x C q C B(0, ) q (2) q m B(0, ).

6 6 This shows that the aveage numbe of codewods in a andomy chosen ba does not depend on the specific code C, but ony on its infomation content m (and the bock ength, which is usuay impicit and cea fom the context). So, we can simpy wite µ m instead of µ C. In the est of this subsection, we show how dense codes ae cosey eated to codes that outpefom the GV bound. This connection is not expicity used in the est of the pape, and it is pesented hee ony fo the pupose of iustating the intuition behind the choice of codes used in the seque. Using function µ m, the definition of the minimum distance (fo codes with infomation content m) guaanteed by the GV bound (see Theoem 5) can be ewitten as d m min{ µ m ( 1) 1}. In paticua, fo any code C[, m, d] q, the expected numbe of codewods in a andom sphee of adius d m is bigge than 1, and thee must exist sphees with mutipe codewods. If the distance of the code d exceeds the GV bound fo codes with the same infomation content (i.e., d > d m), C is (ρ, k)-dense fo some ρ d m/d < 1 and k > 0. Sevea codes ae known fo which the minimum distance exceeds the GV bound: RS codes o, moe geneay, MDS codes, whose distances d exceed d m fo a code ates. RS codes of ate m/ appoaching 1 wi be of paticua inteest. This is due to the fact that the atio d/d m gows with code ate m/ of RS codes and tends to 2 fo high ates. We impicity use this fact in the seque to buid a ρ-dense codes fo any ρ > 1/2. Binay BCH codes, whose distances exceed d m fo vey high code ates appoaching 1. These codes ae not used in this pape, but they ae an impotant famiy of codes that beat the GV bound. AG codes, meeting the TVZ bound, whose distances exceed d m fo most code ates bounded away fom 0 and 1 fo any q. These codes ae used to pove hadness esuts fo asymptoticay good codes, and appoximating the minimum distance up to an additive eo. Remak 7: The eation between dense codes and codes that exceed the GV bound can be given an exact, quantitative chaacteization. In paticua, one can pove that fo any positive intege k and ea ρ, the expected numbe of codewods fom C[, m, d] in a andom sphee of adius ρd is at east q k if and ony if (d m k 1) ρd. B. Dense code famiies fom MDS codes We ae inteested in sequences of (ρ, k)-dense codes with fixed ρ and abitaiy age k. Moeove, paamete k > 0 shoud be poynomiay eated to the bock ength, i.e., θ k fo some θ (0, 1). (Notice that k is a necessay condition because code C contains at most q codewods and we want q k distinct codewods in a ba.) This eation is essentia to aow the constuction of dense codes in time poynomia in k. Sequences of dense codes ae fomay defined beow. Definition 8: A sequence of codes C k [ k, m k, d k ] qk is caed a ρ-dense code famiy if, fo evey k 1, C k is a (ρ, k)-dense code i.e., thee exists a cente v k F k qn such that the ba B(v k, ρd k ) contains qk k codewods. Moeove, we say that C k [ k, m k, d k ] qk is a poynomia code famiy if the bock ength k is poynomiay bounded, i.e., k k c fo some constant c independent of k. In this section we use twice-extended RS codes to buid ρ- dense code famiies fo any ρ > 1/2. Lemma 9: Fo any ɛ (0, 1), and sequence of pime powes q k satisfying k/ɛ 1/ɛ q k O(poy(k)), (3) thee exists a poynomia ρ-dense code famiy G k [ k, m k, d k ] qk with ρ 1/(2(1 ɛ)). Poof: Fix the vaue of ɛ and k and et ρ and q q k be as specified in the emma. Define the adius q ɛ. Notice that, fom the owe bound on the aphabet size, we get ɛ ɛ q ɛ ɛ k/ɛ k. (4) In paticua, since ɛ < 1, we have > k 1 and < q. Since is an intege, it must be 2 q 1. Conside an MDS code G[, m, d] q with q + 1 and d /ρ + 1 meeting the Singeton bound d m + 1 (e.g., twice-extended RS codes, see Section II). Fom the definition of d, we immediatey get ρd ρ( /ρ + 1) >. (5) We wi pove that µ() q ɛ, i.e., the expected numbe of codewods in a andomy chosen ba of adius is at east q ɛ. It immediatey foows fom (4) and (5) that µ( ρd ) µ() q ɛ q k, i.e., the code is (ρ, k)-dense on the aveage. Since MDS codes meet the Singeton bound with equaity, we have m d 1 /ρ /ρ 2(1 ɛ). Theefoe the expected numbe of codewods in a andom sphee satisfies µ m () q m B(0, ) q 2(1 ɛ) B(0, ). (6) We want to bound this quantity. Notice that fo a and we have ( ) 1 ( ) i i. (7) i0 whee we have used that i i fo a and i [0, 1]. A sighty stonge inequaity is obtained as foows: ( ) 1 i0 i i 2 ( + 1) i0 ( ) ( ) ( + 1). (8)

7 7 Moeove, the eade can easiy veify that fo a [2, q] ( + 1) (q + 2) q + 1 2q q + 1. (9) We use (8) and (9) to bound the voume of the sphee B(0, ) as foows: ( ) B(0, ) > (q 1) ( ) ( ) 2q q + 1 (q 1) q + 1 ( ) 2q (1 1q ) ( ) q 2 q ( ) ( 2q 1 ) q + 1 q 2 q (2 ɛ) q (2 ɛ) whee in the ast inequaity we have used q 1 and 1 (q 1)/q 2 (q + 1)/(2q). Combining the bound B(0, ) > q (2 ɛ) with inequaity (6) we get µ m () > q 2(1 ɛ)+(2 ɛ) q ɛ. This poves that ou MDS codes ae (ρ, k)-dense. Moeove, if q k O(poy(k)), then the bock ength q + 1 is aso poynomia in k, and G k is a poynomia ρ-dense famiy of codes. C. Codes ove a fixed aphabet Lemma 9 gives a ρ-dense code famiy G k [ k, m k, d k ] qk fo any fixed ρ > 1/2 with q k O(poy(k)). In the seque, we wish to find a ρ-dense famiy of codes ove some fixed aphabet F q. In ode to keep the aphabet size fixed and sti get abitaiy age k, we take the extension fied F q c and use the MDS codes fom Lemma 9 with aphabet size q c. These codes ae concatenated with equidistant Hadamad codes H[q c 1, c, q c q c 1 ] q to obtain a famiy of dense codes ove fixed aphabet F q. This pocedue can be appied to any dense code as descibed in the foowing emma. Lemma 10: Let C [, m, d ] q c be an abitay code, and et H[q c 1, c, q c q c 1 ] q be the equidistant Hadamad code of size q c. If C is (ρ, k)-dense (aound some cente v), then the concatenated code C C H is (ρ, ck)-dense (aound v H). Poof: Let C [, m, d ] q c be a code and et H[q c 1, c, q c q c 1 ] q be the equidistant Hadamad code of size q c. Define code C as the concatenation of C and H. (See Section II fo detais.) The esuting concatenated code C C H has paametes (q c 1), m cm, d (q c q c 1 ) d. Now assume C is (ρ, k)-dense aound some cente v. Notice that the concatenation function x x H is injective and satisfies wt(x H) (d/d ) wt(x). Theefoe the ba B(x, ρd ) is mapped into ba B(v H, ρd) and the numbe of C -codewods contained in B(v, ρd ) equas the numbe of (C H)-codewods contained in B(v H, ρd). Theefoe C is (ρ, k ) dense fo k og q ((q c ) k ) ck. By inceasing the degee c of the extension fied F q c, we obtain an infinite sequence of q-ay codes G H. Combining Lemma 9 and Lemma 10 we get the foowing poposition. Poposition 11: Fo any ρ > 1/2 and pime powe q, thee exists a poynomia ρ-dense famiy of codes {A k } k1 ove a fixed aphabet Σ F q. Poof: As in Lemma 9, et ɛ > 0 be an abitaiy sma constant and et ρ 1/(2(1 ɛ)). Fo evey k, define c k 1 ɛ og q. Notice that q c k satisfies k ɛ 1/ɛ k q c k O(poy(k)), ɛ so we can invoke Lemma 9 with aphabet size q k q c k and obtain a (ρ, k)-dense code G k [, m, d ] qk. Let A k be the concatenation of G k with Hadamad code H[q c k 1, c k, q c k q ck 1 ] q. By Lemma 10, A k is (ρ, ck)-dense. Moeove, the bock ength of A k is (q c k 1), which is poynomia in k, because both and q c k ae poy(k). This poves that A k is a poynomia ρ-dense famiy. D. Poynomia constuction We poved that ρ-dense famiies of codes exist fo any ρ > 1/2. In this subsection we addess two issues eated to the agoithmic constuction of such codes: Can ρ-dense codes be constucted in poynomia time? I.e., is thee an agoithm that on input k outputs (in time poynomia in k) a inea code which is (ρ, k)-dense? Given a (ρ, k)-dense code, i.e., a code such that some ba B(v, ρd) contains at east q k codewods, can we efficienty find the cente v of a dense sphee? The fist question is easiy answeed: a constuctions descibed in the pevious subsection ae poynomia in k, so the answe to the fist question is yes. The second question is not as simpe and to-date we do not know any deteministic pocedue that efficienty poduces dense codes togethe with a point aound which the code is dense. Howeve, we wi see that, povided the code is dense on the aveage, the cente of the sphee can be efficienty found at east in a pobabiistic sense. We pove the foowing agoithmic vaiant of Poposition 11. Poposition 12: Fo evey pime powe q and ea constant ρ > 1/2, thee exists a pobabiistic agoithm that on input two integes k and s, outputs (in time poynomia in k and s) thee integes, m,, a geneato matix A F m q and a cente v F q (of weight wt(v) ) such that 1) < ρ d(a) 2) with pobabiity at east 1 q s, the ba B(v, ) contains q k o moe codewods. Befoe poving the poposition, we make a few obsevations. The codes descibed in Poposition 11 ae the concatenation of twice-extended RS codes with Hadamad codes, and theefoe they can be expicity constucted in poynomia time. Whie the codes ae descibed expicity, the poof that the code is dense is non constuctive: in Lemma 9 we poved that the aveage numbe of codewods in a andomy chosen sphee is age,

8 8 so sphees containing a age numbe of codewods cetainy exist, but it is not cea how to find a dense cente. At the fist gance, one can ty to educe the entie set of centes F q used in Lemma 9. In paticua, it can be poved that F q can be epaced by any MDS code G that incudes ou oigina MDS code G. Howeve, even with two centes eft, we sti need to count codewods in the two bas to find which is dense indeed. To-date, expicit pocedues fo finding a unique ba ae yet unknown. Theefoe beow we use a pobabiistic appoach and show how to find a dense cente with high pobabiity. Fist, et the cente be chosen unifomy at andom fom the entie space F q. Given the expected numbe of codewods µ, Makov s inequaity yieds P ( B(v, ) G > µ) < 1/, v F q showing that sphees containing at most µ codewods can be found with high pobabiity 1 1/. It tuns out that a simia owe bound P ( B(v, ) G δ µ) δ v B(0,) can be poved if we estict ou choice of v to a unifomy andom eement of B(0, ) ony. This is an instance of a quite genea emma that hods fo any pai of goups G F. 10 In the emma beow, we use mutipicative notation fo goups (G, ) and (F, ). Howeve, to avoid any possibe confusion, we caify in advance that in ou appication G and F wi be goups (G, +) and (F q, +) of codewods with espect to vecto sum opeation. Lemma 13: Let F be a goup, G F a subgoup and B F an abitay subset of F. Given z F, conside the subset Bz {b z b B} and et µ be the aveage size of G Bz as the eement z uns though F. Choose v B 1 {b 1 b B} unifomy at andom. Then fo any δ > 0, P v B 1{ G Bv δµ} δ. Poof: Divide the eements of B into equivaence casses whee u and v ae equivaent if uv 1 G. Then choose v B 1 {b 1 b B} unifomy at andom. If v b 1 is chosen, then G Bv has the same size as the equivaence cass of b. (Notice: the equivaence cass of b is Gb B (G Bv)b.) Since the numbe of equivaence casses is (at most) F / G, the numbe of eements that beong to equivaence casses of size δµ o ess is bounded by δµ F / G (i.e., the maximum numbe of casses times the maximum size of each cass), and the pobabiity that such a cass is seected is at most δµ F /( G B ). The foowing simpe cacuation shows that µ G B / F and theefoe the pobabiity to seect an eement b such that G Bv δµ is at most δ: µ Exp [ G Bz ] z F P {y Bz} z F y G G B. F 10 In fact, it is not necessay to have a goup stuctue on the sets, and the emma can be fomuated in even moe genea settings, but woking with goups make the pesentation simpe. We ae now eady to pove Poposition 12. Poof: Fix some q and ρ > 1/2 and et k, s be the input to the agoithm. We conside the (ρ, k )-dense code A k fom Poposition 11, whee k k + s. This code is the concatenation of a twice extended RS code G[, m, d ] qk fom Lemma 9, and a Hadamad code H[q k 1, c k, q k (1 1/q)] q with bock ength poynomia in k. Theefoe a geneato matix A F m q fo A k can be constucted in time poynomia in k, s. At this point, we instantiate the Lemma 13 with goups F (F q, +), G (G, +), and B B(0, ), whee ρd. Notice that fo any cente z z F q, the set Bz is just the ba B(z, ) of adius centeed in z. Fom the poof of Lemma 9, the aveage size of G Bz (i.e., the expected numbe of codewods in a andom ba when the cente is chosen unifomy at andom fom F q is at east qk+s. Foowing Lemma 13, we choose v F q k unifomy at andom fom B(0, ) B B 1. By Lemma 13 (with δ q s ), we get that B(v, ) contains at east q k codewods with pobabiity 1 q s. Finay, by Lemma 10, we get that the coesponding ba B(v, ) in F q (with adius q k (1 1/q) and cente v v H) contains at east q k codewods fom A. The output of the agoithm is given by a geneating matix fo code A k, the bock ength and infomation content m of this matix, adius and vecto v. E. Mapping dense bas onto fu spaces In the next section we wi use the codewods inside the ba B(v, ) to epesent the soutions to the neaest codewod pobem. In ode to be abe to epesent any possibe soution, we need fist to poject the codewods in B(v, ) to the set of a stings ove F q of some shote ength. This is accompished in the next emma by anothe pobabiistic agument. Given a matix T F k q and a vecto y F q, et T(y) yt denote the inea tansfomation fom F q to F k q. Futhe, et T(Y ) {T(y) y Y }. Lemma 14: Let Y be any fixed subset of F q of size Y q 2k+s. If matix T F k q is chosen unifomy at andom, then with pobabiity at east 1 q s we have T(Y ) F k q. Poof: Choose T F k q unifomy at andom. We want to pove that with vey high pobabiity T(Y ) F k q. Choose a vecto t F k q at andom and define a new function T (y) yt + t. Ceay T (Y ) F k q if and ony if T(Y ) F k q. Notice that the andom vaiabes T (y) (indexed by vecto y Y, and defined by the andom choice of T and t) ae paiwise independent and unifomy distibuted. Theefoe fo any vecto x F k q, T (y) x with pobabiity p q k. Let N x be the numbe of y Y such that T (y) x. By ineaity of expectation and paiwise independence of the T (y) we have Exp [N x ] Y p and Va [N x ] Y (p p 2 ) < Y p. Appying Chebychev s inequaity we get P{N x 0} P{ N x Exp [N x ] Exp [N x ]}

9 9 < Va [N x ] Exp [N x ] 2 1 Y p q (k+s). Theefoe, fo any x F k q, the pobabiity that T (y) x fo evey y Y is at most q (k+s). By union bound, the pobabiity that thee exists some x F k q such that x T (Y ) is at most q s, i.e., with pobabiity at east 1 q s, T (Y ) F k q and theefoe T(Y ) F k q. We combine Poposition 12 and Lemma 14 to buid the gadget needed in the NP-hadness poofs in the foowing section. Lemma 15: Fo any ρ > 1/2 and finite fied F q thee exists a pobabiistic poynomia time agoithm that on input k, s outputs, in time poynomia in k and s, integes, m,, matices A F m q, and T F k q and a vecto v F q (of weight wt(v) ) such that 1) < ρ d(a). 2) with pobabiity at east 1 q s, T(B(v, ) A) F k q, i.e., fo evey x F k q thee exists a y A such that d(y, v) and yt x. Poof: Run the agoithm of Poposition 12 on input k 2k + s + 1 and s s + 1 to obtain integes, m,, a matix A F m q such that < ρd(a) and a cente v F q such that B(v, ) contains at east q 2k+s+1 codewods with pobabiity 1 q s+1. Let Y be the set of a codewods in B(v, ), and choose T F k q unifomy at andom. By Lemma 14, the conditiona pobabiity, given Y q 2k+s+1, that T(Y ) F k q is at east 1 q s+1. Theefoe, the pobabiity that T(Y ) F k q is at most q s+1 + q s+1 q s. IV. HARDNESS OF THE RELATIVELY NEAR CODEWORD PROBLEM In this section we pove that the eativey nea codewod pobem is had to appoximate within any constant facto γ fo a ρ > 1/2. The poof uses the gadget fom Lemma 15. Theoem 16: Fo any ρ > 1/2, γ 1 and any finite fied F q, GAPRNC (ρ ) γ,q is had fo NP unde poynomia RUReductions. Moeove, the eo pobabiity can be made exponentiay sma in a secuity paamete s whie maintaining the eduction poynomia in s. Poof: Fix some finite fied F q. Let ρ be a ea such that Lemma 15 hods tue, et ɛ > 0 be an abitaiy sma positive ea, and γ 1 such that GAPNCP γ,q is NP-had. We pove that GAPRNC (ρ ) γ,q is had fo ρ ρ (1 + ɛ) and γ γ/(2 + 1/ɛ). Since ɛ can be abitaiy sma, Lemma 15 hods fo any ρ > 1/2, and GAPNCP γ,q is NP-had fo any γ 1, this poves the hadness of GAPRNC (ρ ) γ,q fo any γ 1 and ρ > 1/2. The poof is by eduction fom GAPNCP γ,q. Let (C, u, t) be an instance of GAPNCP γ,q with C F k n q. We want to define an instance (C, u, t ) of GAPRNC (ρ ) γ,q such that if (C, u, t) is a YES instance of GAPNCP γ,q, then (C, u, t ) is a YES instance of GAPRNC (ρ ) γ,q with high pobabiity, whie if (C, u, t) is a NO instance of GAPNCP γ,q, then (C, u, t ) is a NO instance of GAPRNC (ρ ) γ,q with pobabiity 1. Notice that the main diffeence between the two pobems is that whie in (C, u, t) the minimum distance d(c) can be abitaiy sma, in (C, u, t ) the minimum distance d(c ) must be eativey age (compaed to eo weight paamete t ). The idea is to embed the oigina code C in a highe dimensiona space to make sue that the new code has age minimum distance. At the same time we want aso to embed taget vecto u in this highe dimensiona space in such a way that the distance of the taget fom the code is oughy peseved. The embedding is easiy pefomed using the gadget fom Lemma 15. Detais foow. On input GAPNCP instance (C, u, t), we invoke Lemma 15 on input k (the infomation content of input code C F k n q ) and secuity paamete s, to find integes, m,, a geneato matix A F m q, a mapping matix T F k q, and a vecto v F q such that: 1) < ρ d(a) 2) T(A B(v, )) F k q with pobabiity at east 1 q s. Conside the inea code ATC F m n q. Notice that a m ows of matix ATC ae codewods of C. (Howeve, ony at most k ae independent.) We define matix C by concatenating 11 b t copies of ATC and a bt ɛ copies of A: C [A,..., A, ATC,..., ATC] (10) }{{}}{{} a b and vecto u as the concatenation of a copies of v and b copies of u: u [v,..., v, u,..., u] (11) }{{}}{{} a b Finay, et t a + bt. The output of the eduction is (C, u, t ). Befoe we can pove that the eduction is coect, we need to bound the quantity a bt and. Using the definition of a and b we get: bt a bt ) ɛ ( bt ɛ ( a bt bt < ɛ + 1) 1 bt ɛ + bt 1 ɛ + ( ) 1 t t ɛ + 1. So, we aways have a bt [ 1 ɛ, 1 ɛ + 1). We can now pove the coectness of the eduction. In ode to conside (C, u, t ) as an instance of GAPRNC (ρ ) γ,q, we fist pove that t < ρ d(c ). Indeed, d(c ) a d(a) > a/ρ and theefoe t ( a + bt d(c < ) a/ρ ρ 1 + bt ) ρ(1 + ɛ) ρ. (12) a Now, assume (C, u, t) is a YES instance, i.e., thee exists x such that d(xc, u) t. Let y za be a codewod in A such 11 Hee the wod concatenation is used to descibe the simpe juxtapposition of matices o vectos, and not the concatenating code constuction of [11] used in Section III.

10 10 that d(y, v) and yt x. We know such a codewod exists with pobabiity at east 1 q s. In such a case, we have d(zc, u ) a d(za, v) + b d(zatc, u) a + bt (13) t poving that (C, u, t ) is a YES instance. Convesey, assume (C, u, t) is a NO instance, i.e., the distance of u fom C is geate than γt. We want to pove that fo a z F m q we have d(zc, u ) > γ t. Indeed, d(zc, u ) b d(z(atc), u) b d(c, u) > b γt γ (2 + 1/ɛ)bt (14) γ ((1 (( + 1/ɛ)bt + bt) > γ a ) ) bt + bt bt γ t poving that (C, u, t ) is a NO instance. (Notice that NO instances get mapped to NO instances with pobabiity 1, as equied.) Remak 17: The eduction given hee is a andomized manyone eduction (o a andomized Kap eduction) which fais with exponentiay sma pobabiity. Howeve it is not a Levin eduction: i.e., given a witness fo a YES instance of the souce of the eduction we do not know how to obtain a witness to YES instances of the taget in poynomia time. The pobem is that given a soution x to the neaest codewod pobem, one has to find a codewod y in the sphee B(v, ) such that yt x. Ou poof ony assets that with high pobabiity such a codewod exists, but it is not known how to find it. This was the case aso fo the Ajtai-Micciancio hadness poof fo the shotest vecto pobem, whee the faiue pobabiity was ony poynomiay sma. As discussed in subsection II-B, hadness unde poynomia RUR-eductions easiy impies the foowing cooay. Cooay 18: Fo any ρ > 1/2, γ 1 and any finite fied F q, GAPRNC (ρ) γ,q is not in RP uness NP RP. Since NP is widey beieved to be diffeent fom RP, Cooay 18 gives evidence that no (pobabiistic) poynomia time agoithm to sove GAPRNC (ρ) γ,q exists. V. HARDNESS OF THE MINIMUM DISTANCE PROBLEM In this section we pove the hadness of appoximating the Minimum Distance Pobem. We fist deive an inappoximabiity esut to within some constant bigge than one by eduction fom GAPRNC (ρ) γ,q. Then we use diect poduct constuctions to ampify the inappoximabiity facto to any constant and to factos 2 og(1 ɛ) n, fo any ɛ > 0. A. Inappoximabiity to within some constant The inappoximabiity of GAPDIST γ,q to within a constant γ (1, 2) immediatey foows fom the hadness of GAPRNC (1/γ) γ,q. Lemma 19: Fo evey γ (1, 2), and evey finite fied F q, GAPDIST γ,q is had fo NP unde poynomia RUR-eductions with exponentiay sma soundness eo. Poof: The poof is by eduction fom GAPRNC γ 1 γ,q. Let (C, u, t) be an instance of GAPRNC γ 1 γ,q with distance d(c) > t/ρ γt. Assume without oss of geneaity that u does not beong to code C. (One can easiy check whethe u C by soving a system of inea equations. If u C then (C, u, t) is a YES instance because d(u, C) 0, and the eduction can output some fixed YES instance of GAPDIST γ,q.) Define the matix C [ C u ]. (15) Assume (C, u, t) is a YES instance of GAPRNCγ,q γ 1, i.e., thee exists an x such that d(xc, u) t. Then, (C, t) is a YES instance of GAPDIST γ,q, since nonzeo vecto xc u beongs to code C and has weight at most t. Convesey, assume (C, u, t) is a NO instance of GAPRNCγ,q γ 1. We pove that any nonzeo vecto y xc+αu of code C has weight above γt. Indeed, if α 0 then y is a nonzeo codewod of C and theefoe has weight wt(y) > γt (since d(c) > γt). On the othe hand, if α 0 then wt(y) wt((α 1 x)c u) > γt as d(u, C) > γt. Hence, d(c ) > γt and (C, t) is a NO instance of GAPDIST γ,q. B. Inappoximabiity to within bigge factos To ampify the hadness esut obtained above, we take the diect poduct of the code with itsef. We fist define diect poducts. Definition 20: Fo i {1, 2}, et A i be a inea code geneated by A i F ki ni q. Then the diect poduct of A 1 and A 2, denoted A 1 A 2 is a code ove F q of bock ength n 1 n 2 and dimension k 1 k 2. Identifying the set F n1n2 q of n 1 n 2 dimensiona vectos with the set F n2 n1 q of a n 2 n 1 matices in the obvious way, the diect poduct A 1 A 2 is convenienty defined as the set of a matices {A T 2 XA 1 X F k2 k1 q } in F n2 n1 q. Notice that a geneating matix fo the poduct code can be easiy computed fom A 1 and A 2, defining a basis codewod (A (i) 1 )T A (j) 2 fo evey ow A (i) 1 of A 1 and ow A (i) 2 of A 2 (whee x T denotes the tanspose of vecto x, and x T y is the standad extena poduct of coumn vecto x T and ow vecto y). Notice that the codewods of A 1 A 2 ae matices whose ows ae codewods of A 1 and coumns ae codewods of A 2. In ou eduction we wi need the foowing fundamenta popety of diect poduct codes. Fo competeness (see aso [17]), we pove it beow. Poposition 21: Fo inea codes A 1 and A 2 of minimum distance d 1 and d 2, thei diect poduct is a inea code of distance d 1 d 2. Poof: Fist, A T 2 XA 1 has at east d 1 d 2 nonzeo enties if X 0. Indeed, conside the matix XA 1 whose ows ae codewods fom A 1. Since this matix is nonzeo, some ow is a nonzeo codewod of weight d 1 o moe. Thus XA 1 has at east d 1 nonzeo coumns. Now conside the matix A T 2 (XA 1 ). At east d 1 coumns of this matix ae nonzeo codewods of A 2. each of weight at east d 2, fo a tota weight of d 1 d 2 o moe.