Department of Mathematical Sciences, Clemson University. Some recent developments in permutation decoding

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1 Department of Mathematcal Scences, Clemson Unversty keyj/ Some recent developments n permutaton decodng J. D. Key keyj@ces.clemson.edu 1/32 P

2 Abstract The method of permutaton decodng was frst developed by MacWllams n the early 60 s and can be used when a lnear code has a suffcently large automorphsm group to ensure the exstence of a set of automorphsms, called a PD-set, that has some specfed propertes. Ths talk wll descrbe some recent developments n fndng PD-sets for codes defned through the row-span over fnte felds of ncdence matrces of desgns or adjacency matrces of regular graphs, snce these codes have many propertes that can be deduced from the combnatoral propertes of the desgns or graphs, and often have a great deal of symmetry and large automorphsm groups. May 8, /32 P

3 Codng theory termnology A lnear code s a subspace of a fnte-dmensonal vector space over a fnte feld. (All codes are lnear n ths talk.) The weght of a vector s the number of non-zero coordnate entres. If a code has smallest non-zero weght d then the code can correct up to d 1 2 errors by nearest-neghbour decodng. A code C s [n, k, d] q f t s over F q and of length n, dmenson k, and mnmum weght d. A generator matrx for the code s a k n matrx made up of a bass for C. The dual code C s the orthogonal under the standard nner product (, ),.e. C = {v F n (v, c) = 0 for all c C}. A check matrx for C s a generator matrx H for C. 3/32 P

4 Codng theory termnology contnued Two lnear codes of the same length and over the same feld are somorphc f they can be obtaned from one another by permutng the coordnate postons. An automorphsm of a code C s an somorphsm from C to C. Any code s somorphc to a code wth generator matrx n standard form,.e. the form [I k A]; a check matrx then s gven by [ A T I n k ]. The frst k coordnates are the nformaton symbols and the last n k coordnates are the check symbols. 4/32 P

5 Permutaton decodng Permutaton decodng was frst developed by Jesse MacWllams [Mac64] followng also Prange [Pra62]. It can be used when a code has suffcently many automorphsms to ensure the exstence of a set of automorphsms called a PD-set. Early work was mostly on cyclc codes and the Golay codes. We extend the defnton of PD-sets to s-pd-sets for s-error-correcton [KMM05]: Defnton 1 If C s a t-error-correctng code wth nformaton set I and check set C, then a PD-set for C s a set S of automorphsms of C whch s such that every t-set of coordnate postons s moved by at least one member of S nto the check postons C. For s t an s-pd-set s a set S of automorphsms of C whch s such that every s-set of coordnate postons s moved by at least one member of S nto C. Specfcally, f I = {1,..., k} are the nformaton postons and C = {k +1,..., n} the check postons, then every s-tuple from {1,..., n} can be moved by some element of S nto C. 5/32 P

6 Algorthm for permutaton decodng C s a q-ary t-error-correctng [n, k, d] q code; d = 2t + 1 or 2t + 2. k n generator matrx for C: G = [I k A]. Any k-tuple v s encoded as vg. The frst k columns are the nformaton symbols, the last n k are check symbols. (n k) n check matrx for C: H = [ A T I n k ]. S = {g 1,..., g m } s a PD-set for C, wrtten n some chosen order. Suppose x s sent and y s receved and at most t errors occur: for = 1,..., m, compute yg and the syndrome s = H(yg ) T untl an s found such that the weght of s s t or less; 6/32 P

7 Algorthm for permutaton decodng C s a q-ary t-error-correctng [n, k, d] q code; d = 2t + 1 or 2t + 2. k n generator matrx for C: G = [I k A]. Any k-tuple v s encoded as vg. The frst k columns are the nformaton symbols, the last n k are check symbols. (n k) n check matrx for C: H = [ A T I n k ]. S = {g 1,..., g m } s a PD-set for C, wrtten n some chosen order. Suppose x s sent and y s receved and at most t errors occur: for = 1,..., m, compute yg and the syndrome s = H(yg ) T untl an s found such that the weght of s s t or less; f u = u 1 u 2... u k are the nformaton symbols of yg, compute the codeword c = ug; 6/32 P

8 Algorthm for permutaton decodng C s a q-ary t-error-correctng [n, k, d] q code; d = 2t + 1 or 2t + 2. k n generator matrx for C: G = [I k A]. Any k-tuple v s encoded as vg. The frst k columns are the nformaton symbols, the last n k are check symbols. (n k) n check matrx for C: H = [ A T I n k ]. S = {g 1,..., g m } s a PD-set for C, wrtten n some chosen order. Suppose x s sent and y s receved and at most t errors occur: for = 1,..., m, compute yg and the syndrome s = H(yg ) T untl an s found such that the weght of s s t or less; f u = u 1 u 2... u k are the nformaton symbols of yg, compute the codeword c = ug; decode y as cg 1. 6/32 P

9 Why permutaton decodng works Result 1 Let C be an [n, k, d] q t-error-correctng code. Suppose H s a check matrx for C n standard form,.e. such that I n k s n the redundancy postons. Let y = c+e be a vector, where c C and e has weght t. Then the nformaton symbols n y are correct f and only f the weght of the syndrome Hy T of y s t. 7/32 P

10 Tme complexty A smple argument yelds that the worst-case tme complexty for the decodng algorthm usng an s-pd-set of sze m on a code of length n and dmenson k s O(nkm). So small PD-sets are desrable. Further, snce the algorthm uses an orderng of the PD-set, good choces of the orderng of the elements can reduce the complexty. For example: fnd an m-pd-set S m for each 0 m t such that S 0 < S 1... < S t and arrange the PD-set S n ths order: S 0 (S 1 S 0 ) (S 2 S 1 )... S t S t 1. (Usually take S 0 = {d}). 8/32 P

11 Mnmum sze for a PD-set Countng shows that there s a mnmum sze a PD-set can have; most the sets known have sze larger than ths mnmum. The followng s due to Gordon [Gor82], usng a result of Schönhem [Sch64]: Result 2 If S s a PD-set for a t-error-correctng [n, k, d] q code C, and r = n k, then n n 1 n t + 1 S r r 1 r t + 1 (Proof n Huffman [Huf98].) Ths result can be adapted to s-pd-sets for s t by replacng t by s n the formula. Example: The bnary extended Golay code, parameters [24, 12, 8], has n = 24, r = 12 and t = 3, so S = 14 and PD-sets of ths sze has been found (see Gordon [Gor82] and Wolfmann [Wol83]). 9/32 P

12 Desgn theory background An ncdence structure D = (P, B, I), wth pont set P, block set B and ncdence I s a t-(v, k, λ) desgn, f P = v, every block B B s ncdent wth precsely k ponts, and every t dstnct ponts are together ncdent wth precsely λ blocks. Takng t 1, the number of blocks ncdent wth a gven pont s constant for the desgn, called the replcaton number for D. If B = b then bk = vr. E.g. A 2-(n 2 + n + 1, n + 1, 1) s a projectve plane of order n, where n = p e s a prme power (n all known cases); a 2 (16, 6, 2) s a bplane. The code C F of the desgn D over the fnte feld F s the space spanned by the ncdence vectors of the blocks over F,.e. the row span over F of a b v ncdence matrx, a 0-1 matrx wth k 1 s n every row and r 1 s n every column: see [AK92, AK96]. Smlarly, the code of a regular undrected graph Γ over a fnte feld F s the row span over F of an adjacency matrx for Γ. Ths matrx has k 1 s n every row and column, where k s the valency of the graph. 10/32 P

13 Fndng PD-sets Frst we need an nformaton set. These are not known n general; further dfferent nformaton sets wll yeld dfferent possbltes for PD-sets. For symmetrc desgns (e.g. projectve planes), a bass of ncdence vectors of blocks wll yeld a correspondng nformaton set, by dualty. Ths lnks to the queston of fndng bases of mnmum-weght vectors n the geometrc case, agan somethng not known n general. For planes, Moorhouse [Moo91] or Blokhus and Moorhouse [BM95] gve bases n the prme-order case. Recently a convenent nformaton set for the desgns of ponts and hyperplanes of prme order was found n [KMM06] (I ll get back to ths.) NOTE: Magma [BC94] has been a great help n lookng at small cases to get the general dea of what to mght hold for the general case and nfnte classes of codes. 11/32 P

14 Classes of codes havng s-pd-sets If Aut(C) s k-transtve then Aut(C) tself s a k-pd-set, n whch case we attempt to fnd smaller sets; exstence of a k-pd-set s not nvarant under code somorphsm; codes from the row span over a fnte feld of an ncdence matrx of a desgn or geometry, or from an adjacency matrx of a graph; usng Result 2 t follows that many classes of desgns and graphs where the mnmum-weght and automorphsm group are known, cannot have PD-sets for full error-correcton for length beyond some bound; for these we look for s-pdsets wth 2 s < d 1 2 : e.g. fnte planes, Paley graphs; for some classes of regular and sem-regular graphs wth large automorphsm groups, PD-sets exst for all lengths: e.g. bnary codes of trangular graphs, lattce graphs, lne graphs of complete mult-partte graphs. 12/32 P

15 Some nfnte classes of codes havng PD-sets In all of these, sutable nformaton sets had to be found. 1. Trangular graphs For any n, the trangular graph T (n) s the lne graph of the complete graph K n, and s strongly regular. (The vertces are the ( n 2) 2-sets, wth two vertces beng adjacent f they ntersect: ths s n the class of unform subset graphs.) The row span over F 2 of an adjacency matrx gves codes: [ n(n 1) 2, n 1, n 1] 2 for n odd and [ n(n 1) 2, n 2, 2(n 1)] 2 for n even where n 5. The automorphsm group s S n actng naturally (apart from n = 5) and get PD-sets of sze n for n odd and n 2 2n + 2 for n even, by [KMR04b]. (The computatonal complexty of the decodng by ths method may be qute low, of the order n 1.5 f the elements of the PD-set are approprately ordered.) 13/32 P

16 2. Graphs on trples Defne three graphs wth vertex set the subsets of sze three of a set of sze n and adjacency accordng to the sze of the ntersecton of the 3-subsets. these codes are n [KMR04a]. Propertes of Agan S n n ts natural acton s the automorphsm group. The ternary codes of these graphs are also of nterest. If C s the bnary code n the case of adjacency f the 3-subsets ntersect n two elements, then the dual C s a [ ( ) ( n 3, n 1 ) 2, n 2]2 code and a PD-set of n 3 can be found by [KMR]. W. Fsh (Cape Town) s workng on bnary codes from unform subset graphs n general (odd graphs, Johnson graphs, Knesner graphs, etc.) 14/32 P

17 3. Lattce graphs The (square) lattce graph L 2 (n) s the lne graph of the complete bpartte graph K n,n, and s strongly regular. The row span over F 2 of an adjacency matrx gves codes: [n 2, 2(n 1), 2(n 1)] 2 for n 5 wth S n S 2 as automorphsm group, and PD-sets of sze n 2 n S n S n were found n [KSc]. (The lower bound from Result 2 s O(n).) A smlar result holds for the rectangular lattce graph L 2 (m, n), m < n: the codes are [mn, m + n 2, 2m] 2 for m + n even, [mn, m + n 1, m] 2 for m + n odd. PD-sets of sze m and m + n, respectvely, n S m S n can be found. [KSa]. More generally for the lne graphs of mult-partte graphs, wth automorphsm group S n1 S n2... S nm : [KSb]. 15/32 P

18 Complexty of permutaton decodng The followng can be used to order the PD-set for the bnary code of the square lattce graph. Proposton 1 For the [n 2, 2(n 1), 2(n 1)] 2 code from the row span of an adjacency matrx of the lattce graph L 2 (n), usng nformaton set {(, n) 2 n 1} {(n, ) 1 n}, for 0 k t = n 2, S k = {((, n), (j, n)) n k, j n} s a k-pd-set, where (n, n) denotes the dentty permutaton n S n. Thus orderng the elements of the PD-set as S 0, S 1 S 0, S 2 S 1,..., S n 2 S n 3 wll result n a PD-set where, f s t = n 2 errors occur then the search through the PD-set need only go as far as s th block of elements. Snce the probablty of less errors s hghest, ths wll reduce the tme complexty. 16/32 P

19 Proposton 2 If C s the bnary code formed by the row space over F 2 of an adjacency matrx for the rectangular lattce graph L 2 (m, n) for 2 m < n, then C s [mn, m + n 2, 2m] 2 for m + n even; [mn, m + n 1, m] 2 for m + n odd. The set I = {(, n) 1 m} {(m, ) 1 n 1} s an nformaton set for m+n odd, and I\{(1, n)} s an nformaton set for m + n even. The sets of automorphsms S s = {((, m), (, n)) 1 2s} {d} for m + n odd; S s = {((, m), (j, n)) 1 m, 1 j s} {d} for m + n even are s error correctng PD-sets for any 0 s t errors. A study of the complexty of the algorthm for some algebrac geometry codes s gve n [Joy05]. 17/32 P

20 Some nfnte classes of codes only havng partal PD-sets 1. Fnte planes If q = p e where p s prme, the code of the desarguesan projectve plane of order q has parameters: C = [q 2 + q + 1, ( p(p+1) 2 ) e + 1, q + 1] p. For the affne plane the code s [q 2, ( p(p+1) 2 ) e, q] p. Smlarly, the desgns formed from ponts and subspaces of dmenson r, for some r, n projectve or affne space, have GRM codes and the parameters are known. The codes are subfeld subcodes of the generalzed Reed-Muller codes, and the automorphsm groups are the sem-lnear groups and doubly transtve. 18/32 P

21 Thus 2-PD-sets always exst but the bound for full error-correcton of Result 2 s greater than the sze of the group (see [KMM05]) as q gets large. For example, n the projectve desarguesan case when: q = p prme and p > 103; q = 2 e and e > 12; q = 3 e and e > 6; q = 5 e and e > 4; q = 7 e and e > 3; q = 11 e and e > 2; q = 13 e and e > 2; q = p e for p > 13 and e > 1. Smlar results hold for the affne and dual cases, n all of the desgns. 19/32 P

22 Informaton sets for generalzed Reed-Muller codes R Fq (ρ, m) = x 1 1 x 2 2 x m m 0 k q 1, for 1 k m, m k ρ. k=1 s the ρ th -order generalzed Reed-Muller code R Fq (ρ, m), of length q m over the feld F q. In [KMM06] we found nformaton sets for these codes: Theorem 3 Let V = F m q, where q = p t and p s a prme, and F q = {α 0,..., α q 1 }. Then m I = {(α 1,..., α m ) k ν, 0 k q 1} k=1 s an nformaton set for R Fq (ν, m). If q = p s a prme, I = {( 1,..., m ) k F p, 1 k m, m k ν} k=1 s an nformaton set for R Fp (ν, m), by takng α k = k. 20/32 P

23 Examples to llustrate the theorem q = m = x 0 1 x0 2 = 1 [0,0] x 0 1 x1 2 [0,1] x 0 1 x2 2 [0,2] x 1 1 x0 2 [1,0] x 1 1 x1 2 [1,1] x 2 1 x0 2 [2,0] Fgure 1: R Fq (ρ, m) = R F3 (2, 2) = [9, 6, 3] 3 B = {x 1 1 x k 2, }. 21/32 P

24 Proposton 4 If C = C p (P G m,m 1 (F p )), where p s a prme and m 2, then, usng homogeneous coordnates, the ncdence vectors of the set {(1, a 1,..., a m ) a F p, m a p 1} {(0,..., 0, 1) } =1 of hyperplanes form a bass for C. Smlarly, a bass of hyperplanes for C p (AG m,m 1 (F p )) for m 2, p prme s the set of ncdence vectors of the hyperplanes wth equaton m a X = p 1 =1 wth m a p 1, =1 where a F p for 1 m, and not all the a are 0, along wth the hyperplane wth equaton X m = 0. 22/32 P

25 Example A bass of mnmum-weght vectors for C 3 (P G 2,1 (F 3 )) (0, 0, 1) (1, 0, 0) (1, 0, 1) (1, 0, 2) (1, 1, 0) (1, 1, 1) (1, 2, 0) Fgure 2: C 3 (P G 2,1 (F 3 )) 23/32 P

26 Example A bass of mnmum-weght vectors for R F3 (2, 2) = C 3 (AG 2,1 (F 3 )) X 2 = X 2 = X 2 = X 1 = X 1 + X 2 = X 1 = Fgure 3: R F3 (2, 2) = C 3 (AG 2,1 (F 3 )) Compare wth the generator matrx usng the polynomal bass 1. 24/32 P

27 Small 2-PD-sets n prme-order planes 2-PD-sets exst for any nformaton set (snce the group s 2-transtve); for prme order, usng a Moorhouse [Moo91] bass, 2-PD-sets of 37 elements for the [p 2, ( ) p+1 2, p]p codes of the desarguesan affne planes of any prme order p and 2-PD-sets of 43 elements for the [p 2 +p+1, ( ) p , p+1]p codes of the desarguesan projectve planes of any prme order p were constructed n [KMM05]. Also 3-PD-sets for the code and the dual code n the affne prme case of szes 2p 2 (p 1) and p 2, respectvely, were found. Other orders q and other codes from geometres yeld smlar results. 25/32 P

28 2.Ponts and lnes n 3-space Theorem 5 [KMM] Let D be the 2-(p 3, p, 1) desgn AG 3,1 (F p ) of ponts and lnes n the affne space AG 3 (F p ), where p s a prme, and let C = R Fp (2(p 1), 3) be the p-ary code of D. Then C s a [p 3, 1 6 p(5p2 + 1), p] p code wth nformaton set I = {( 1, 2, 3 ) k F p, 1 k 3, 3 k 2(p 1)}. (1) k=1 Let T s the translaton group, D the group of nvertble dagonal 3 3 matrces, and Z the group of scalar matrces, and, for each d F p wth d 0, let δ d be the assocated dlataton. Usng ths nformaton set, for p 5 there exsts d F p such that C has an 2-PD-set of the form T T δ d of sze 2p 3, and for p 7 T D s a 3-PD-set for C of sze p 3 (p 1) 3. (In fact d = p 1 2 wll be sutable for the 2-PD-set.) Note: These codes have hgh rate /32 P

29 3. Paley graphs If n s a prme power wth n 1(mod 4), the Paley graph,p (n), has F n as vertex set and two vertces x and y are adjacent f and only f x y s a non-zero square n F n. The row span over a feld F p of an adjacency matrx gves an nterestng code (quadratc resdue codes) f and only f p s a square n F n. For any σ Aut(F n ) and a, b F n wth a a non-zero square, the group of maps τ a,b,σ : x ax σ + b s the automorphsm group of the code, and for n 1697 and prme or n 1849 and a square, PD-sets cannot exst snce the bound of Result 2 s bgger than the order of the group (usng the square root bound for the mnmum weght, and the actual mnmum weght q + 1 when n = q 2 and q s a prme power). 27/32 P

30 For the case where n s prme and n 1 (mod 8), the code of P (n) over F p s C = [n, n 1 2, d] p where d n, (the square-root bound) for p any prme dvdng n 1 4. C has a 2-PD-set of sze 6 by [KL04]. (The automorphsm group s not 2-transtve.) For the dual code n ths case, a 2-PD-set of sze 10 for all n was found. Further results n [Lm05]. 28/32 P

31 References [AK92] [AK96] E. F. Assmus, Jr and J. D. Key. Desgns and ther Codes. Cambrdge: Cambrdge Unversty Press, Cambrdge Tracts n Mathematcs, Vol. 103 (Second prntng wth correctons, 1993). E. F. Assmus, Jr. and J. D. Key. Desgns and codes: an update. Des. Codes Cryptogr., 9:7 27, [BC94] Web Bosma and John Cannon. Handbook of Magma Functons. Department of Mathematcs, Unversty of Sydney, November [BM95] [Gor82] Aart Blokhus and G. Erc Moorhouse. Some p-ranks related to orthogonal spaces. J. Algebrac Combn., 4: , D. M. Gordon. Mnmal permutaton sets for decodng the bnary Golay codes. IEEE Trans. Inform. Theory, 28: , [Huf98] W. Cary Huffman. Codes and groups. In V. S. Pless and W. C. Huff- 29/32 P

32 man, edtors, Handbook of Codng Theory, pages Amsterdam: Elsever, Volume 2, Part 2, Chapter 17. [Joy05] [KL04] Davd Joyner. Conjectural permutaton decodng of some AG codes. ACM SIGSAM Bulletn, 39, No.1, March. J. D. Key and J. Lmbupasrporn. Permutaton decodng of codes from Paley graphs. Congr. Numer., 170: , [KMM] J. D. Key, T. P. McDonough, and V. C. Mavron. Partal permutaton decodng of codes from affne geometry desgns. Submtted. [KMM05] J. D. Key, T. P. McDonough, and V. C. Mavron. Partal permutaton decodng of codes from fnte planes. European J. Combn., 26: , [KMM06] J. D. Key, T. P. McDonough, and V. C. Mavron. Informaton sets and partal permutaton decodng of codes from fnte geometres. Fnte Felds Appl., 12: , [KMR] J. D. Key, J. Moor, and B. G. Rodrgues. Bnary codes from graphs on trples and permutaton decodng. Ars Combn. To appear. 30/32 P

33 [KMR04a] J. D. Key, J. Moor, and B. G. Rodrgues. Bnary codes from graphs on trples. Dscrete Math., 282/1-3: , [KMR04b] J. D. Key, J. Moor, and B. G. Rodrgues. Permutaton decodng for bnary codes from trangular graphs. European J. Combn., 25: , [KSa] J. D. Key and P. Senevratne. Bnary codes from rectangular lattce graphs and permutaton decodng. European J. Combn.,To appear. [KSb] J. D. Key and P. Senevratne. Codes from the lne graphs of complete multpartte graphs and PD-sets. Submtted. [KSc] [Lm05] [Mac64] J. D. Key and P. Senevratne. Permutaton decodng of bnary codes from lattce graphs. Dscrete Math. (Specal ssue dedcated to J. Seberry), To appear. J. Lmbupasrporn. Partal permutaton decodng for codes from desgns and fnte geometres. PhD thess, Clemson Unversty, F. J. MacWllams. Permutaton decodng of systematc codes. Bell System Tech. J., 43: , /32 P

34 [Moo91] [Pra62] G. Erc Moorhouse. Bruck nets, codes, and characters of loops. Des. Codes Cryptogr., 1:7 29, E. Prange. The use of nformaton sets n decodng cyclc codes. IRE Trans., IT-8:5 9, [Sch64] J. Schönhem. On coverngs. Pacfc J. Math., 14: , [Wol83] J. Wolfmann. A permutaton decodng of the (24,12,8) Golay code. IEEE Trans. Inform. Theory, 29: , /32 P