Cyclic Low-Density MDS Array Codes

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1 Cyclic Low-Desity MDS Aay Codes 1 Yuval Cassuto Depatmet of Electical Egieeig Califoia Istitute of Techology Pasadea, CA 91125, U.S.A. ycassuto@paadise.caltech.edu Jehoshua Buck Depatmet of Electical Egieeig Califoia Istitute of Techology Pasadea, CA 91125, U.S.A. buck@paadise.caltech.edu Abstact We costuct two ifiite families of low desity MDS aay codes which ae also cyclic. Oe of these families icludes the fist such sub-family with edudacy paamete > 2. The two costuctios have diffeet algebaic fomulatios, though they both have the same idiect stuctue. Fist MDS codes that ae ot cyclic ae costucted ad the by applyig a cetai mappig to thei paity check matices, o-equivalet cyclic codes with the same distace ad desity popeties ae obtaied. Usig the same poof techiques, a thid ifiite family of quasi-cyclic codes ca be costucted. I. INTRODUCTION MDS (maximum distace sepaable) codes ove lage symbol alphabets ae ubiquitous i data stoage applicatios. Beig MDS, they offe the maximum potectio agaist device failues fo a give amout of edudacy. Aay codes ae oe type of such codes that is vey useful to dyamic high-speed stoage applicatios as they ejoy low-complexity decodig algoithms ove small fields as well as low update complexity whe small chages ae applied to the stoed cotet. That is i cotast to the family of Reed-Solomo codes that i geeal has oe of these favoable popeties. Examples of eplacemets costuctios that yield aay codes with these popeties ca be foud i [1],[2],[3],[4],[5] ad [6] (patial list). I this pape we wish to popose codes of this type that ae also cyclic. As a example we examie the followig code defied o a 2 6 aay a 0 a 1 a 2 a 3 a 4 a a a 2 a 3 a 4 0 a 1 a 2 a 1 a 2 a 3 a 3 a 4 a 5 a 4 a 5 a 0 a 5 a 0 This code has 6 ifomatio bits a 0,..., a 5 all of which ca be ecoveed fom ay set of 3 colums that i total have 6 bits. Hece the code is MDS. Howeve, the focus of this pape is a diffeet popety of this sample code; its cyclicity. To covice oeself that the code is cyclic, we obseve that all the idices i a colum ca be obtaied by addig oe (modulo 6) to the idices i the colum to its (cyclic) left. Thus ay shift of the ifomatio bits ow esults i a idetical shift i the paity bits ow. Beyod theoetical iteest, cyclic aay codes whose othe popeties match those of thei best-kow o-cyclic coutepats, offe sigificat pactical advatages. Usig cyclic 1 This wok was suppoted i pat by the Caltech Lee Cete fo Advaced Netwokig ad by NSF gat ANI a 1 aay codes fo high-speed stoage applicatios ca educe the implemetatio cost of the codes. This impoved costpefomace is thaks to savigs i time ad space esouces eeded fo ecodig ad decodig ad also by allowig may useful opeatios o the aay to be caied out usig simple egula cicuits. Aothe advatage is havig a uifom desig fo the idividual stoage uits that implemet the code aay. Examples of beefits of cyclic aay codes ae povided i sectio III. The peviously oly kow family of low-desity MDS cyclic codes is the code with edudacy = 2 that was poposed i [6]. I this pape we peset two families of cyclic MDS aay codes. The fist, give i sectio IV, is defied o aays with dimesios p 1 (p 1), whee is the code edudacy ad p is a pime. It icludes the fist family of cyclic MDS aay codes with > 2. Its costuctio builds upo a kow costuctio fo o-cyclic codes (fist appeaed i [1], late exteded i [7], ad i [5]) by fist shoteig the codes ad the explicitly povidig a class of mappigs of paity check matix locatios fom the oigial shoteed codes to the cyclic codes. The secod costuctio, give i sectio V, poposes a ew o-cyclic code family o aays with dimesios (p 1) (p 1) ad = 2, the poves its MDS popety ad similaly shows how to map the codes to cyclic codes. The fist pat of that costuctio (the ocyclic pat) ca be alteately peseted usig gaph theoetic costuctio tools: pefect 1-factoizatios of complete bipatite gaphs, ad thus it is a geealizatio of the method poposed i [5] that uses factoizatios of complete ui-patite gaphs. Combiig the poof techiques fom the two codes costucted i this pape, a thid family of lowest desity quasi-cyclic MDS codes ca be costucted. These codes have dimesios (p 1) 2(p 1) ad = 2. Due to space limitatios, we omit the pesetatio of these codes to allow a geate focus o the poof techiques. To make the costuctios cleae, a example fo each oe is povided followig its fomal desciptio. II. DEFINITIONS A liea aay code (C) F of dimesios b ove F = F q is a liea subspace of the vecto space F b. The dual code (C) F is the ull-space of (C) F ove F. To defie the miimum distace of a aay code we egad it as a code ove the alphabet F b, whee F b deotes legth b vectos ove F. The the miimum distace is simply the miimum Hammig distace of the legth code ove F b. Note that though the

2 code symbols ca be egaded as elemets i the fiite field F q b, we do ot assume lieaity ove this field. (C) F ca be specified by eithe its Paity-check matix H of size N p b o its Geeato matix G of size (b N p ) b. A Paitycheck (o Geeato) matix is called systematic if it has N p (o b N p ) ot ecessaily adjacet colums that whe stacked togethe fom the idetity matix I Np (o I b Np ), espectively. Give a systematic H matix o G matix (oe ca be easily obtaied fom the othe), the b symbols of the b aay ca be patitioed ito N p paity symbols ad b N p ifomatio symbols. Defie the desity of the code N(G) as the aveage umbe of o-zeos i a ow of G, b N p, whee N(M) is the umbe of o-zeos i a matix M. Whe H is systematic a alteative expessio fo the desity is 1 + N(H) Np b N p. We call a code (C) F lowest desity if its desity equals its miimum distace (the miimum distace is a obvious lowe boud o the desity [4]). We call a family of codes low-desity if the desity of the codes is O(1). If b N p ad the miimum distace d equals Np b +1 the the code is called maximum distace sepaable (MDS) with edudacy = Np b. Thoughout the pape [s, t] deotes the set {x Z : s x t}. To simplify the pesetatio of the costuctios i the pape, we itoduce aothe stuctue that defies a code whe, as is the situatio hee, the paity check matix has elemets i {0, 1}. Give a paity check matix H, defie the idex aay A H to be a b aay of subsets of [0, N p 1]. Idex aays ca be similaly defied fo geeato matices as well, but these do ot appea i this pape. h l deotes the l th colum of H ad h l (x) the x th elemet of h l, x [0, N p 1]. The set i locatio i, j of A H cotais the elemets {x : h i+bj (x) = 1}. If H is systematic, A H has N p subsets of size 1. Note that A H has the same dimesios as the code aay ad its sets specify the ecodigs of the bits of (C) F. As a example we take a ( = 6, b = 3, N p = 6) systematic code ad povide i figue 1 a geeato matix G ad a paity check matix H with its idex aay A H. III. CYCLIC ARRAY CODES The codes we heeafte costuct ae codes of legth ove F b which ae cyclic but ot liea. I this sectio we wish to discuss such codes i geeal, povidig coditios fo a code to be cyclic ad highlightig the potetial beefits of usig them. Oe way to chaacteize cyclic aay codes is as cyclic goup codes ove the diect-poduct goup of the additive goup of F. Aothe is to view them as legth b liea b-quasi-cyclic codes. Fo the most pat, the latte view will pove moe useful sice the costuctios below ae ot explicit goup theoetic oes. I fact, the desciptio of aay codes usig idex aays we chose hee was used i [8] to descibe quasi-cyclic code costuctios. We stat off with the basic defiitio of cyclic codes. Defiitio 1: The code C ove F b is cyclic if s = (s 0, s 1,..., s 1 ) C s = (s 1, s 0,..., s 2 ) C ad s i F b. The followig popositio is a staight fowad geealizatio of a folkloe fact about liea cyclic codes. Let S be the matix G = b H = Fig. 1. 4, 5 5, 0 0, 1 1, 2 2, 3 3, 4 1, 3 2, 4 3, 5 4, 0 5, 1 0, 2 b G,H ad A H fo a sample = 6, b = 3, N p = 6 code b Np Np that cyclically ight-shifts the colums of H, b times. If H is viewed as cocateatio of matices of size N p b, H = [H 0 H 1 H 1 ] the HS = [H 1 H 0 H 2 ]. Popositio 1: 1) The code (C) F with a paity check matix H is cyclic if ad oly if thee exists a ivetible N p N p matix L such that HS = LH. 2) The code (C) F with a geeato matix G is cyclic if ad oly if thee exists a ivetible (b N p ) (b N p ) matix L such that GS = L G. Coollay 1: If (C) F is cyclic, so is (C) F. While popositio 1 is ot uique to aay codes, fo a class of aay codes that ae soo defied, a stoge ecessay ad sufficiet coditio fo cyclicity ca be poved. A paity check matix of a systematic aay code is called egula if N p ad the subset J l [1, b] of the colums of H l that epeset paity bits is the same fo evey l. I figue 1 H is egula sice the fist colum of evey N p b sub-matix coespods to a paity bit. The vecto that has oe i locatio i ad zeos else whee is deoted by e i. Theoem 1: A code (C) F with a egula paity check matix H is cyclic if ad oly if HS = L s H whee L s is the N p N p

3 pemutatio matix that pefoms Np dowwad cyclic shift of the ows of H. Coollay 2: A code give as a egula idex aay A H is cyclic if ad oly if addig Np modulo N p to the elemets of the sets of A H yields a cyclic shift of A H. A. Meits of cyclic codes Oe dimesioal liea cyclic codes ae kow to povide geat advatages such as succict epesetatios ad efficiet ecodig ad decodig. Cyclic aay codes cay simila advatages whe used to potect stoed data. A cyclic aay code ca be specified by povidig oly the b Np subsets of the set [0, N p 1], coespodig to the fist colum of the code aay. This makes seachig fo cyclic codes a moe tactable opeatio (the esults of sectio IV povide a example whee estictig a seach to cyclic codes may ot compomise the popeties of the codes foud). As fo decodig, a cyclic code ca povide a facto savigs i memoy used to decode easues. A geeic (ad sometimes the best kow) way to decode easues i aay codes is stoig a decodig matix fo evey combiatio of colum easues. This matix is the ivese of the cocateatio of the sub-matices (of size N p b) of H that coespod to the eased colums. With a cyclic code evey such easue combiatios have cyclically equivalet decodig sub-matices ad eed oly a sigle stoed decodig matix. May othe opeatios ca be caied out usig simple cicuits whe the code is cyclic. As examples we ca take sydome computatio fo eo decodig ad update opeatios such as bit,ow o colum updates. Usig cyclic codes ca also pove pactically appealig fo stoage applicatios sice its symmety allows the stoage devices to have idetical desigs, compaed to, i geeal, a specialized desig fo each uit depedig o its idex i the code wod. IV. CYCLIC LOWEST-DENSITY MDS CODES WITH = p 1, b = p 1 Let be a diviso of p 1, ad p a odd pime. Let α be a elemet i F p of ode ad β be a elemet i F p of ode p 1. α ad β defie a patitio of F p to cosets of its multiplicative subgoup of ode plus a set that cotais oly the zeo elemet. Except fo the zeo set, all sets ae of cadiality ad thee ae p 1 such sets. C 1 = {0} C i = {β i, β i α,..., β i α 1 } (1) whee 0 i < p 1. The sets C i ae used i [4] ad [7] to costuct (o-cyclic) lowest desity MDS codes with edudacy. The costuctio theei is a geealizatio of the = 2 costuctio of [1]. I [7], this costuctio was poved to povide lowest desity MDS codes fo a wide age of paametes. Whe F has chaacteistic 2, MDS codes ae obtaied fo = 3 ad = 4 wheeve 2 is pimitive i F p. Fo lage chaacteistics, codes with additioal values wee show to be MDS. Fo completeess we peset a costuctio fo codes which ae shoteed vesios of the codes theei. Late i the sectio we show that fo evey code costucted i this mae, thee exists a class of mappigs fom locatios i the paity check matix of the code to those of a diffeet (o equivalet) code that is cyclic. Ude these mappigs, the ew cyclic codes iheit the distace ad desity popeties fom [1] ad [7], so the poposed codes ejoy the cyclicity popety while ot compomisig the optimality of thei acestos. Bette eadability i mid ad with a slight abuse of otatio, opeatios o sets deote elemet-wise opeatios o the cotet of the sets. Specifically, if x + l z is used to deote x + l (mod z), the S + l z deotes the set that is obtaied by addig l to the elemets of S modulo z; also deote S z S + 0 z. Similaly, pemutatios ad aithmetic opeatios o sets epeset the coespodig opeatios o thei elemets. Fo evey 0 m < p 1 defie I m = {i : x C i + m p, 0 x < p 1}. It is obvious that fo evey m, I m = p 1 sice fo evey taslatio m of the sets C i oly oe set cotais the elemet p 1. Deote the j th elemet of I m by I m (j), whee idices i I m ae odeed lexicogaphically. The code C is defied via a idex aay A H. The set at locatio (j, m) [0, p 1 1] [0, p 2], i A H is C i + m p, i = I m (j). The code C is a shoteed vesio of the code costucted i [4],[7]. Note that because of the estictio i I m, Theoem 1 implies that C is ot a cyclic code. To defie the cyclic code C ( fo cyclic), we deive alteative costuctig sets D i fom C i, as descibed below. The pemutatio ψ : [0, p 2] [0, p 2] is defied to be ψ(x) = β x 1 (mod p). We also defie the ivese of ψ, ψ 1 (y) = log β (y+1). Fo i I 0, defie D i = ψ 1 (C i ). C is similaly defied though its idex aay A H as follows. The j, l set of A H, (j, l) [0, p 1 1] [0, p 2], is D i +l p 1, ad ow i = I 0 (j) (cf. i = I m (j) i the defiitio of C). The fact that fo evey l taslatios of the same sets D i ae take, togethe with Coollay 2 poves the followig popositio. Popositio 2: The code C is cyclic. Theoem 2: C ad C have the same edudacy, miimum distace ad desity. poof: We show that C ca be obtaied fom C (ad also vice vesa) by a distace-pesevig bijectio betwee ows ad colums of H to those of H (i aay codes colum pemutatios of the paity check matix, i geeal, do ot peseve distace). To efe to a elemet x i the set at locatio (j, l) i a idex aay A M we use the tuple (x, j, l, M). The afoemetioed bijectio is give by showig that A H is obtaied fom A H by mappig (x, j, l, H ) (ψ(x), j, m, H). The mappig x ψ(x) epesets pemutig the ows of the paity check matix ad the mappig (j, l) (j, m) epesets pemutig colums of the paity check matix. The mappig (j, l) (j, m) has a special popety that it oly eodes colums of the idex aay ad eodes sets withi its colums. Hece the mappig (x, j, l, H ) (ψ(x), j, m, H) peseves both the miimum distace ad desity of the code. Moe cocetely, we eed to show that fo evey l [0, p 2] thee exists a m [0, p 2] such that evey i = I 0 (j) has a coespodig t = I m (j ) that togethe satisfy ψ[ D i + l p 1 ] = C t + m p

4 is MDS (has miimum colum weight 3) we show that fo evey l 1, l 2, the matix that is obtaied by the juxtapositio [H l1 H l2 ] is osigula (hece thee ae o weight 2 wods i the code). As a emide, H li is the N p b = 2(p 1) (p 1) matix that coespods to colum l i of the idex aay A H, i the way descibed i sectio II. To do that we esot to some additioal defiitios. ψ[ D i +l p 1 ] = ψ[ ψ 1 [C i ]+l p 1 ] = β log β ( Ci+1 p 1)+l 1 p Let E be a m m matix with eties e ij {0, 1}. Defie a Sice D 1 +l p 1 cosists of the sigle elemet l ad C 1 + m p cosists of the sigle elemet m, the iteges l ad m have to satisfy m = ψ(l). The we ewite the above coditio as ψ[ D i + l p 1 ] = C t + ψ(l) p = β l C i + β l 1 p = C i+l p 1 + ψ(l) p A. Example: cyclic MDS code with p=7, =6, b=3, =2 I F 7 pick α = 6, β = 3 that satisfy od(α) = = 2, od(β) = p 1 = 6. These α ad β defie the followig patitio of F 7 ito sets C i accodig to (1). C 1 = {0}, C 0 = {1, 6}, C 1 = {3, 4}, C 2 = {2, 5} Takig the sets C i + m 7 to be the sets of A H i colum m, leavig out the paticula set i that colum that cotais the elemet 6, we get 3, 4 0, 2 1, 3 4, 2 5, 3 2, 1 2, 5 5, 4 4, 0 1, 5 0, 1 3, 0 The pemutatios ψ ad ψ 1 witte explicitly ae ψ [0, 1, 2, 3, 4, 5] [0, 2, 1, 5, 3, 4] ad [0, 1, 2, 3, 4, 5] [0, 2, 1, 4, 5, 3]. ψ 1 actig o the aay A H yields ψ 1 (A H ) = , 5 0, 1 2, 4 5, 1 3, 4 1, 2 1, 3 3, 5 5, 0 2, 3 0, 2 4, 0 ψ 1 which afte eodeig of colums ad sets withi colums esults i the cyclic code geeated by D i + l 6, i I 0 = { 1, 1, 2}. A H = 4, 5 5, 0 0, 1 1, 2 2, 3 3, 4 1, 3 2, 4 3, 5 4, 0 5, 1 0, 2 V. CYCLIC LOW-DENSITY MDS CODES WITH = p 1, b = p 1, = 2 Let p be a odd pime. Defie the odeed pais C i, i [0, p 2] to be C i = (a i, c p 1 i ) (2) Fo a odeed pai P = (a j, c m ) defie P (l) = (a j, c m+l p ). The pais C (l) i ae fist used to defie a o-systematic, ocyclic MDS code B. To defie the code B, we agai use the idex aay A H, but ow, fo coveiece of pesetatio, the set [0, N p 1] = {0, 1,..., 2(p 1) 1} is mapped to the set {a 0, c 0, a 1, c 1,..., a p 2, c p 2 }. The set at locatio (0, l), l [0, p 2], i A H is {a l } ad the set at locatio (j, l) [1, p 2] [0, p 2], is {C (l) j+l p 1 }. Note that sice the idex of c i C (l) i is icemeted modulo p while the idex of a is icemeted modulo p 1, B is ot cyclic. To pove that B sigleto colum to be a colum with a sigle 1. Two ows i 1, i 2 ae called 2 coected if thee is a colum with exactly two 1s i locatios i 1, i 2. Defie a potudig ow to be a ow with a 1 i a sigleto colum. A ow is called oble, fo easos that will soo become clea, if it is potudig o if it is 2 coected to aothe oble ow. It is ot had to see that if a set of ow vectos i E is liealy depedet it does ot iclude ay oble ows. Usig the above, the followig lemma is faily easy to pove. Lemma 1: If i a matix E all of the ows ae oble, the E is osigula. Now take E = [H l1 H l2 ]. E has two sigleto colums (ad coespodig two potudig ows) defied by the sets {a l1 }, {a l2 }, ad 2(p 2) 2 coectios defied by the sets {C (l1) i }, {C (l2) i }. We wat to show that all the ows of E ae oble by aguig that each ow a 0, c 0, a 1, c 1,..., a p 2, c p 2 is coected to a potudig ow by a chai of 2 coectios. Give the two colums l 1 = l, l 2 = l +, 1 < p 1, the chais of oble ows statig fom a l, a l+ ae, espectively a l c 1 a l c 2 1 a l 2 c t 1 a l+ c 1 a l+2 c 2 1 a l+3 c s 1 This follows sice by costuctio (2), i colums l ad l + the a ad c idices sum to l 1 ad l + 1, espectively. If s + t < p the the chais ae disjoit ad eithe iclude epetitios. Othewise thee would be s, t : s + t < p such that t 1 = s 1 (mod p) o l t = l + s (mod p) ad both ae impossible sice gcd(, p) = 1. Theefoe, the fist ad secod chais ecessaily temiate whe t 1 = l (mod p) ad s 1 = l + (mod p) espectively. These coespod to c l ad c l+ that ae abset fom colums l ad l + espectively. To show that all 2(p 1) ow idices appea i the chais we subtact the two equatios ad get (t + s + 1) = 0 (mod p) ad thus t+s = p 1. We coclude that all ows ae oble ad E = [H l1 H l2 ] is o-sigula fo evey l 1, l 2. A. Cyclic code Let β be a pimitive elemet i F p. The pemutatio ψ : [0, p 2] [0, p 2] is oce agai defied to be ψ(x) = β x 1 (mod p). The ivese pemutatio ψ 1 is the ψ 1 (y) = log β (y + 1). Fo otatioal coveiece we use φ(a i ),φ(c i ) to deote a φ(i), c φ(i) espectively, whee φ is a abitay pemutatio ad also a i + l,c i + l fo a i+l,c i+l espectively. Defie the odeed pais D j, j [1, p 2] to be D j = ( ψ 1 (a j ), ψ 1 (c p 1 j ) ) (3)

5 The code B has a idex aay A H whose set i locatio (0, l), l [0, p 2] is {a l } ad the set i locatio (j, l) [1, p 2] [0, p 2] is D j +l p 1. To use Coollay 2 to pove the cyclicity of B, we map {a 0, c 0, a 1, c 1,..., a p 2, c p 2 } back to [0, 2p 3] ad obseve that addig 2 modulo 2(p 1) to the sets mapped fom D j + l p 1 yields a cyclic shift of A H. Theoem 3: B ad B have the same edudacy, miimum distace ad desity. poof: We show, similaly to the poof of Theoem 2, that B ca be obtaied fom B (ad also vice vesa) by a distacepesevig bijectio betwee ows ad colums of H ad those of H. Specifically, we pove the claim that fo evey j [1, p 2] thee exists a i [0, p 2] such that ψ( D j +l p 1 ) = C (ψ(l)) i. Note that as metioed befoe, the a idices ad c idices of C (m) i sum to m 1 (mod p). The poof simplifies thaks to the followig two obsevatios. Fist, the elemets of D j + l p 1 ae distict fo diffeet j. Secod, thee ae exactly p 2 pais s, t [0, p 2] that give s + t = m 1 (mod p) fo evey m [0, p 2] (this is ot tue fo m = p 1 whee we have p 1 such pais). Cosequetly, povig that the a ad c idices of ψ( D j +l p 1 ), sum to ψ(l) 1 fo each j establishes that these sets ae ideed C (ψ(l)) i. So povig the followig suffices. ψ(ψ 1 (j) + l) + ψ(ψ 1 (p 1 j) + l) = ψ(l) 1 (mod p) Povig the above is the staight fowad ψ(ψ 1 (j) + l) + ψ(ψ 1 (p 1 j) + l) = = β log β (j+1)+l 1 + β log β (p 1 j+1)+l 1 = = β l (j p j) 2 = β l 1 1 = ψ(l) 1 B. Systematic paity-check matix The paity check matix obtaied fo B (ad similaly fo B) ealie i the sectio is ot systematic. The bits c i do ot appea i it as sigleto colums. No-systematic paity check matices ae udesiable sice they do ot offe the simple ecodig allowed by systematic oes. Moeove, whe the paity check matix is systematic, oe ca easily use it to extact the desity of the code. We ow deive a systematic, cyclic paity check matix fom the o-systematic cyclic oe of sectio V-A by settig c i = c i a i fo i [0, p 2]. Sice the a ad c idices of C (m) i sum to m 1 {p 1} [0, p 3], fo evey m thee exists a pai C (m) i of the fom C (m) i = (a (m 1)/2, c (m 1)/2 ). The sets D i ae obtaied fom C i by pemutatio so fo each m, oe of the sets D j + m p 1 is of the fom (a ψ 1 ((m 1)/2), c ψ 1 ((m 1)/2)). Thus the tasfomatio c i = c i a i makes the paity check matix systematic. The umbe of oes i each colum of the o-systematic pat of the modified paity check matix is 3, theefoe the desity is 4. C. Example: cyclic MDS code with p=5, =4, b=4, =2 Fo p = 5 the sets C i ae C 0 = (a 0, c 4 ), C 1 = (a 1, c 3 ), C 2 = (a 2, c 2 ), C 3 = (a 3, c 1 ) Note that the idices of a ad c i each of the sets C i sum to p 1 = 4. Takig the set C (l) j+l 4 to be the (j, l) set of A H, a 0 a 1 a 2 a 3 a 1, c 3 a 2, c 3 a 3, c 3 a 0, c 2 a 2, c 2 a 3, c 2 a 0, c 1 a 1, c 1 a 3, c 1 a 0, c 0 a 1, c 0 a 2, c 0 Now pick a pimitive elemet i F 5 β = 2 ad use it to defie the pemutatios ψ ad ψ 1 ψ, [0, 1, 2, 3] [0, 1, 3, 2], [0, 1, 2, 3] ψ 1 [0, 1, 3, 2]. The pemutatio ψ 1 actig o the sets of A H yields ψ 1 (A H ) = a 0 a 1 a 3 a 2 a 1, c 2 a 3, c 2 a 2, c 2 a 0, c 3 a 3, c 3 a 2, c 3 a 0, c 1 a 1, c 1 a 2, c 1 a 0, c 0 a 1, c 0 a 3, c 0 which afte eodeig of colums ad sets withi colums esults i the cyclic code geeated by D j + l 4, A H = a 0 a 1 a 2 a 3 a 1, c 2 a 2, c 3 a 3, c 0 a 0, c 1 a 3, c 3 a 0, c 0 a 1, c 1 a 2, c 2 a 2, c 1 a 3, c 2 a 0, c 3 a 1, c 0 VI. CONCLUSION A impotat atifact of the two code families peseted hee is that they take thei MDS popety fom thei o-cyclic pees. A diect poof of thei MDS popety is still missig, ad if foud, it may eable the costuctio of ew families of cyclic codes. This optimistic view is suppoted by compute seaches that eveal MDS lowest-desity cyclic codes with paametes that ae ot coveed by the kow families of ocyclic codes. REFERENCES [1] G.V Zaitsev, V.A Ziov ev, ad N.V Semakov. Miimum-check-desity codes fo coectig bytes of eos, easues, o defects. Poblems Ifom. Tasm., 19: , [2] M. Blaum, J. Bady, J. Buck, ad J. Meo. EVENODD: a efficiet scheme fo toleatig double disk failues i RAID achitectues. IEEE Tasactios o Computes, 44(2): , [3] M. Blaum, J. Buck, ad A. Vady. MDS aay codes with idepedet paity symbols. IEEE-Tas-IT, 42(2): , [4] M. Blaum ad R.M Roth. O lowest desity MDS codes. IEEE-Tas-IT, 45(1):46 59, [5] L. Xu, V. Bohossia, J. Buck, ad D.G Wage. Low-desity MDS codes ad factos of complete gaphs. IEEE-Tas-IT, 45(6): , [6] L. Xu ad J. Buck. X-code: MDS aay codes with optimal ecodig. IEEE-Tas-IT, 45(1): , [7] E. Louido ad R.M Roth. Lowest-desity MDS codes ove extesio alphabets. Techio CS Techical epot, available at pdf. [8] R.L Towsed ad E.J Weldo J. Self-othogoal quasi-cyclic codes. IEEE-Tas-IT, 13(2): , 1967.