Partial permutation decoding for MacDonald codes

Transcription

1 Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics Texas A&M University-Commerce 2600 South Nea Street, Commerce TX 75428, U.S.A. January 28, 2016 Abstract [ q n q u We show how to find s-pd-sets of the minima size s+1 for the q 1, n, qn 1 q ]q u 1 MacDonad q-ary codes C n,u (q) where n 3 and 1 u n 1. The construction of [6] can be used and gives s-pd-sets for s up to the bound (n u)(q 1) 1, of effective use for u sma; for u n 2 an aternative construction is given that appies up to a bound that depends on the maximum size of a set of vectors in V u (F q ) with each pair of vectors distance at east 3 apart. qn u 1 Keywords: MacDonad codes, simpex codes, PD-sets Mathematics Subject Cassifications (2010): 05B05, 94B05 1 Introduction The MacDonad codes, introduced by MacDonad [12] for binary codes, with the definition extended to q-ary codes in [2, 15], are punctured simpex codes, of ength qn q u for any n and [ q n q u 1 u n 1. They have parameters q 1, n, qn 1 q ]q u 1 and are 2-weight codes with the non-zero words of weight q n 1 q u 1 and q n 1. Foowing [2], we denote the codes by C n,u (q). In [6] permutation decoding for the simpex codes was considered and s-pd-sets of the minima size s+1 were found for s up to some arge bound. Since the efficiency of the decoding depends on the size of the PD-set, those of minima size as determined by the Gordon-Schönheim bound (see Resut 1) are the best ones to obtain. The ideas behind the estabishment of these PD-sets for the simpex codes can be appied to the MacDonad codes, and we obtain here some simiar resuts. The automorphism group used for the permutation decoding is a subgroup of the genera inear group and thus described as n n matrices. A genera construction of suitabe matrices aows us to prove a genera theorem: Theorem 1. For n 3 and 1 u n 1, q a prime power, et C n,u (q) denote the [ q n q u q 1, n, qn 1 q u 1 ]q MacDonad q-code. Then C n,u(q) has s-pd-sets of size s + 1 keyj@cemson.edu Padmapani.Seneviratne@tamuc.edu q 1 1

2 2 BACKGROUND AND TERMINOLOGY 2 for a s such that 1 s f n u (F q ) = q n u 1 (n u)(q 1) 1; for n 4 for a s such that 1 s D 1 where D is a set of vectors in V u (F q ) which is such that any two members of D are distance at east 3 apart. Such s-pd-sets can be expicity given in terms of matrices in GL n (F q ). The construction for the first bound is given in Proposition 2 and Coroary 1, using Resut 3, and for the second in Proposition 3 and Coroary 2. The s-pd-sets are a nested in the sense that if 1 r s then any subset of the set of size r + 1 wi correct r errors. Unike in the simpex case, these sets from Coroary 1 are too sma, and therefore correct too few errors, if u is arge. In that case one uses the second construction (of Proposition 3), and particuar exampes of these arise from any q-ary code of ength u and minimum weight at east 3: for exampe, the Hamming codes. Other exampes exist by finding sets of s + 1 vectors in V u (F q ) such that the distance between any two vectors in the set is at east 3. We describe our notation and give some background definitions in Section 2 and Section 3, and prove the resuts on the s-pd-sets of size s + 1 in Section 4. Incuded as Section 6 is an appendix giving some tabes showing the best size of s for sets of size s + 1 using our methods, for 4 n 10, 1 u n 1, and q = 2, 3, 4, 5. Magma [3, 5] was used for any computations, and in particuar to find the sets in the case where u is arge. 2 Background and terminoogy The notation for codes is standard and can be found in [1]. The codes here are a inear codes, and the notation [n, k, d] q wi be used for a q-ary code C of ength n, dimension k, and minimum weight d, where the weight wt(v) of a vector v is the number of non-zero coordinate entries. The distance, d(u, v), between two vectors u, v is wt(u v), i.e. the number of coordinate paces in which they differ. A generator matrix for an [n, k, d] q code C is a k n matrix whose rows form a basis for C, and the dua code C is the orthogona under the standard inner product (, ), i.e. C = {v F n (v, c) = 0 for a c C}. A check matrix for C is a generator matrix for C. Foowing [1, Definition 2.2.3], two inear codes over the same fied are caed equivaent if each can be obtained from the other by permuting the coordinate positions and mutipying each coordinate by a non-zero fied eement. The codes wi be said to be isomorphic if a permutation of the coordinate positions suffices to take one to the other. Generay, an automorphism of a code C is a code equivaence from C to C, and the set of a these gives the automorphism group of the code, written Aut(C) or MAut(C) (foowing [8, Chapter 7,Section 1.3]), since they are given by monomia matrices, and we do not consider here the more genera case that incudes fied automorphisms, or the Gaois groups. If ony permutations of the coordinate positions are aowed then the group of permutation automorphisms is, again foowing [8, Chapter 7, Section 1.3], caed the permutation automorphism group, written PAut(C). Any code is isomorphic to a code with generator matrix in so-caed standard form, i.e. the form [I k A]; a check matrix then is given by [ A T I n k ]. The set of the first k coordinate positions in the standard form is caed an information set for the code, and the set of the ast n k coordinate positions is the corresponding check set. Permutation decoding was deveoped by MacWiiams [13] and invoves finding a set of automorphisms of a code caed a PD-set. The method is described fuy in MacWiiams and

3 3 MACDONALD CODES 3 Soane [14, Chapter 16, p. 513] and Huffman [8, Section 8]. In [9] and [11] the definition of PD-sets was extended to that of s-pd-sets for s-error-correction: Definition 1. If C is a t-error-correcting code with information set I and check set C, then a PD-set for C is a set S of automorphisms of C which is such that every t-set of coordinate positions is moved by at east one member of S into the check positions C. For s t an s-pd-set is a set S of automorphisms of C which is such that every s-set of coordinate positions is moved by at east one member of S into C. The agorithm for permutation decoding is as foows: we have a t-error-correcting [n, k, d] q code C with check matrix H in standard form. Thus the generator matrix G = [I k A] and H = [ A T I n k ], for some A, and the first k coordinate positions correspond to the information symbos. Any vector v of ength k is encoded as vg. Suppose x is sent and y is received and at most t errors occur. Let S = {g 1,..., g s } be the PD-set. Compute the syndromes H(yg i ) T for i = 1,..., s unti an i is found such that the weight of this vector is t or ess. Compute the codeword c that has the same information symbos as yg i and decode y as cgi 1. Notice that this agorithm actuay uses the PD-set as a sequence. Thus it is expedient to index the eements of the set S by the set {1, 2,..., S } so that eements that wi correct a sma number of errors occur first. Thus if nested s-pd-sets are found for a 1 < s t then we can order S as foows: find an s-pd-set S s for each 0 s t such that S 0 S 1... S t and arrange the PD-set S as a sequence in this order: S = [S 0, (S 1 S 0 ), (S 2 S 1 ),..., (S t S t 1 )]. (Usuay one takes S 0 = {id}.) There is a bound on the minimum size that a PD-set S may have, due to Gordon [7], from a formua due to Schönheim [16], and quoted and proved in [8]: Resut 1. If S is a PD-set for a t-error-correcting [n, k, d] q code C, and r = n k, then n n 1 n t + 1 S (1) r r 1 r t + 1 This resut can be adapted to s-pd-sets for s t by repacing t by s in the formua. Note that since r < n, the innermost term for any s is 2, and since there are s terms in the formua, it foows that the size of an s-pd-set is at east s + 1. We wi use the foowing from [10, Lemma 7]: Resut 2. Let C be a code with minimum distance d, I an information set, C the corresponding check set and P = I C. Let A be an automorphism group of C, and n the maximum of O I / O, where O is an A-orbit. If s = min( 1 d 1 n 1, 2 ), then A is an s-pd-set for C. 3 MacDonad codes The q-ary simpex code S n (F q ), for any prime-power q, is a q-ary code with generator matrix having for coumns any set of qn 1 q 1 representatives of the distinct 1-dimensiona subspaces of V n (F q ), i.e. the points of the projective space P G n 1 (F q ): see, for exampe, [1, Section 2.5]. Thus for q > 2 the actua code depends on the representatives chosen, but the codes are of course a equivaent. It foows that S n (F q ) is a [ qn 1 q 1, n, qn 1 ] q code and a the non-zero

4 3 MACDONALD CODES 4 words have weight q n 1 : see [1, Section 2.5]. The coordinate positions are abeed by the projective points P G n 1 (F q ). The automorphism group is isomorphic to ΓL n (q), as shown in [8, Section 7]. The MacDonad codes were introduced by MacDonad [12] for p = 2 and in [15] (see aso [2]) for any q, and are simpex codes punctured in a particuar way. Thus for any number u such that 1 u n 1, if e i denotes the i th basis eement of the standard basis for the vector space V n (F q ) = F n q, et U = e i n u + 1 i n, i.e. a the vectors with the first n u positions equa to 0. The qu 1 q 1 coordinate positions for the vectors in U are removed to produce a code of ength qn q u q 1, with coordinate positions abeed by the points of P G n 1(F q ) that are not in the projective space P G u 1 (F q ). The code sti has dimension n but now it is a two-weight code, with words of weight q n 1 and q n 1 q u 1, as can easiy be seen. We denote these codes by C n,u (q), foowing [2]. If q > 2 then different choices of the representatives of the projective points (coumns of the generator matrix) wi produce different codes, but the codes wi be equivaent. Thus for a q, n, u where 1 u n 1, C n,u (q) is a [ q n q u ] q 1, n, qn 1 q u 1 q-ary code with non-zero codewords of weight q n 1 or q n 1 q u 1. It is cear that if I n,u = {e i 1 i n u} {e 1 + e i n u + 1 i n} = {w i 1 i n}, (2) then the corresponding projective points wi form an information set for C n,u (q) for any n, u, q, where we write w i = e i for 1 i n u, and w i = e 1 + e i for n u + 1 i n. As in [6] we wi write our vectors as rows and have an n n matrix A act on v V n (F q )\{0} (i.e. v P G n 1 (F q )) as va. However, if q > 2, it coud happen that va is not the chosen representative of the projective point va. In such cases automorphisms of the code C are sti produced, but do not correspond to permutations of the code, i.e. they are in MAut(C) but not in PAut(C). In this case the action of the n n matrix A on the code is as described in [6, Coroary 5], and invoves using a generator matrix G, in standard form, for the code, and finding a monomia matrix M such that G T A = MG T. Then the action on the code is given by c cm T for any codeword c, and permutation decoding may sti be used. In the binary case we can aways just use the matrix itsef. For simpicity, we wi choose the coumns of the defining generating matrix to have first entry 1. In this way any suitabe matrix in upper trianguar form with 1 s on the diagona wi directy give a member of PAut(C). Let M m,n (F q ) denote the m n matrices over F q. We take n 3 to avoid trivia cases or codes that woud not be used for permutation decoding. Proposition 1. Let n 3, q a prime power, 1 u n 1, C n,u (q) the MacDonad code, [ A C K = { 0 B ] A GL n u (F q ), B GL u (F q ), C M n u,u (F q )}, q and H = {M M G, B = I u }.

5 4 S-PD-SETS OF SIZE S Then K preserves C n,u (q), and H acts transitivey on the coordinate positions of C n,u (q). Furthermore, H is an s-pd-set for C n,u (q), for any information set, where s = min( qn q u qn 1 n(q 1) 1, q u 1 1 ). 2 Proof: Ceary any such matrix wi map a vector of the form (0,..., 0, a n u+1,..., a n ) to one of the same form so the vectors corresponding to the coordinate positions are preserved. To show the group H is transitive on the coordinate positions we can show that e 1 can map to any vector v from the coordinate set by taking a matrix with v as the first row. Thus any member of the coordinate set can be mapped to any other by some automorphism of the code induced by a member of H. To show that H is an s-pd-set as stated, use Resut 2. Note: The permutation group PAut(C n,u (q)) may not be transitive for q > 2. [ ] 1 c Exampe 1. For u = n 1, H = { c V 0 I n 1 (F q )}. Thus H = q n 1 and the n 1 group H wi be a s-pd-set for C n,n 1 (q) for n 3, where s = qn q n 1 n(q 1) 1 = qn 1 n 1, since this is smaer than, or equa to, qn 1 q n s-pd-sets of size s + 1 We can use the resuts of [6] to obtain s-pd-sets of size s + 1 for the MacDonad codes using automorphisms from the matrix group GL n (F q ) acting on the coordinate positions. Reca that we noted after giving Resut 1 (the Gordon-Schönheim bound) that for correcting s errors the PD-set must have size at east s + 1. We consider the binary case, since the ideas can be extended to the q-ary case as we, as in [6]. The coordinate positions for C n,u (2) are the vectors from V n (F 2 ) \ U, where U = e i n u + 1 i n. The information set is I n,u as given in Equation (2). Proposition 2. For n 3, 1 u n 1, et C n,u (2) be the [2 n 2 u, n, 2 n 1 2 u 1 ] 2 MacDonad code with information set I n,u and check set C n,u. For fixed k 1, et [ ] Ni 0 P k = {M i = 0 i k} 0 I u where N i GL n u (F 2 ), be a set of k + 1 matrices in GL n (F 2 ) such that no two matrices Ni 1 and Nj 1 for i j have a row in common. Then P k is a k-pd-set of k +1 eements for C n,u (2) with information set I n,u. Furthermore, any subset of P k of size s + 1 where 1 s k is an s-pd-set for C n,u (2). [ ] Hi 0 Conversey, if R k = {L i = 0 i k} K is a k-pd-set for C n,u (2) then no two matrices H 1 i and H 1 j [ Proof: Note first that Mi 1 N 1 = i 0 0 I u two matrices N 1 i and N 1 j 0 I u for i j have a row in common. ]. Suppose P k = {M i 0 i k} as shown, and no for i j have a row in common. Let T = {v 1,..., v k } be a set of k

6 4 S-PD-SETS OF SIZE S distinct vectors in V n (F 2 ) \ U. Suppose that we cannot map T into C n,u by any eement of P k. Then for each i such that 0 i k, there is a v j T, for 1 j k, such that v j M i I n,u. Since there are k + 1 vaues of i but ony k of j we must have v j M i and v j M, for some j, and i, both in I n,u. Suppose v j M i = w r and v j M = w t ; then v j = w r Mi 1 = w t M 1. There are three cases to consider: Case (1): r, t n u. Then w r = e r and w t = e t, so the r th row of Mi 1 is the t th row of M 1, contradicting our assumption. Case (2): r n u, t > n u. Then w r = e r and w t = e 1 + e t. Thus w r Mi 1 is the r th row of Mi 1, which has the ast u digits equa to 0, and w t M 1 = (e 1 + e t )M 1, i.e. the sum of the first row of M 1 and the t th row, and this does not have 0 s in the ast u digits. Case (3): r > n u, t > n u. Then w r = e 1 + e r and w t = e 1 + e t. Thus w r Mi 1 is the sum of the first and r th rows of Mi 1, and w t M 1 is the sum of the first and t th rows of M 1. This is again a contradiction, since the first rows of Mi 1 and M 1 are different. Ceary the above argument works for any s with 1 s k. The proof of the remaining statement is virtuay the same as that given in [6, Proposition 1]. The proof of the proposition extends easiy to C n,u (q) for any q, noting that we take the rows of the matrices Ni 1 to be normaized and the action of the matrix on the code might not be a permutation automorphism, as described in Section 3. The construction of sets of matrices as required here is discussed in [6] and the reader is referred to that paper and references there to see how this can be achieved. In particuar, the number q n 1 f n (F q ) = 1 (3) n(q 1) for n 1, was introduced, it being the maximum vaue of s for which an s-pd-set of size s + 1 for the simpex codes can exist: see [6, Lemma 2]. This aso aows for construction of the matrices N i with the properties required in Proposition 2; a specific construction is given in [6, Lemma 5]: Resut 3. For n 2, q 2 a prime power, et K = F q n and et ζ be a primitive eement of K. For 0 i f n (F q ), if B i = {ζ j+in 0 j n 1}, then {B i 0 i f n (F q )} is a set of f n (F q ) + 1 mutuay disjoint bases for V n (F q ). For our purposes we need matrices of size (n u), but the same construction hods. Each matrix Ni 1 has for its rows the fied eements of a basis B i expressed as vectors in V n u (F q ), and normaized. Thus we have: Coroary 1. For n 3 and 1 u n 1, q a prime power, the MacDonad code C n,u (q) has s-pd-sets of size s + 1 for a s such that 1 s f n u (F q ) = 1. q n u 1 (n u)(q 1) Exampe 2. If q = 3 and n u = 3 we have f 3 (F 3 ) = 3, and we use Resut 3 to construct, with the computationa hep [ of Magma ] [3, 5], four normaized matrices, I 3 = N0 1 and Ni 1 Ni 0 for 1 i 3, such that { 0 i 3} forms a 3-PD-set of minima size 4 for 0 I u C n,n 3 (3), with the usua information set. Thus we took F 27 with primitive poynomia x 3 +2x+1, primitive root ζ, and B 0 = {1, ζ, ζ 2 }, B 1 = ζ 3 B 0, B 2 = ζ 6 B 0, B 3 = ζ 9 B 0. Expressing these as vectors in V 3 (F 3 ) and normaising

7 4 S-PD-SETS OF SIZE S gives the matrices: I 3, N 1 1 = , N 1 2 = , N 1 3 = , with inverses I 3, N 1 = , N 2 = , N 3 = These matrices may now be used for decoding C n,n 3 (3) for any n 4, recaing of course that simpe matrix mutipication wi not give the automorphism here, so that the corresponding monomia matrices as described in Section 3 wi need to be constructed to compete the decoding. If n u is sma it is not possibe to find suitabe matrices N i, and in particuar if u = n 1, or n 2, uness the fied is arge in the atter case. A different approach is needed here. Proposition 3. For n 4, 3 u n 1, q any prime, et C n,u (q) be the [ qn q u q 1, n, qn 1 q u 1 ] q q-ary MacDonad code with information set I n,u and check set C n,u. For fixed k 1, et [ ] In u B P k = {M i = i 0 i k} 0 I u where B i M n u,u (F q ), with rows r i, for 1 n u, such that for any i j: wt(r i, r j,m ) 2 for a, m and wt(r i,1 r j,1 ) 3. Then P k is a k-pd-set of k+1 eements for C n,u (q) with information set I n,u. Furthermore, any subset of P k of size s + 1 where 1 s k is an s-pd-set for C n,u (q). [ ] Proof: We proceed in the same way as in Proposition 2. Note first that Mi 1 In u B = i 0 I u for 0 i k so the same set of conditions appy to the rows of B i. Let T = {v 1,..., v k } be a set of k distinct vectors in V n (F q ) \ U. Suppose that we cannot map T into C n,u by any eement of P k. Then for each i such that 0 i k, there is a v j T, for 1 j k, such that v j M i I n,u. Since there are k + 1 vaues of i but ony k of j we must have v j M i and v j M, for some j, and i, both in I n,u. Suppose v j M i = w r and v j M = w t ; then v j = w r Mi 1 = w t M 1. There are three cases to consider: Case (1): r, t n u. Then w r = e r, w t = e t. Since the r th and t th rows of B i and B are distinct, this is not possibe. Case (2): r n u and t > n u. So w r = e 1 and w t = e 1 + e t. Thus w r Mi 1 = w t M 1 becomes the r th row of Mi 1 equa to the sum of the first and t th rows of M 1, which impies that r i,r r,1 has weight 1, which is not possibe. Case (3): r, t > n u. Then we have (e 1 + e r )Mi 1 = (e 1 + e t )M 1, and thus r i,1 r,1 is a vector of weight at most 2, again not possibe. The ast statement foows as before.

8 5 CONCLUSION 8 Note: (1) Since wt(r i,1 r j,1 ) 3 for i j, we can, without oss of generaity, take B i to be the (n u) u matrix a of whose rows are r i,1. We can thus search for k + 1 vectors in V u (F q ) such that any two are distance at east 3 apart. (2) If u = 2 we must use the method of Proposition 2. This wi yied some sets as ong as n > 3. For n = 3 and u = 2 the q-ary code is a [q 2, 3, q 2 q] q code and other methods coud be empoyed. (3) Since the matrices in this construction are a upper trianguar, with 1 s on the diagona, they are in the permutation group of the code so the action on the coordinate positions is by direct mutipication. Using the matrices B i as defined in the first note above, we have: Coroary 2. Let n 4, 3 u n 1, any q. Let D be a set of vectors in V u (F q ) which is such that any two members of D are distance at east 3 apart. Then the D matrices B i, for 1 i D, where B i is an (n u) u matrix with a rows the i th vector from D, wi give a set of D automorphisms of C n,u (q) that form a ( D 1)-PD-set using the information set I n,u. Note: A subspace of minimum weight at east 3 coud serve as the set D. Methods for constructing codes with a prescribed minimum distance are given in [4] for the binary case, and [17] for the q > 2 case. Exampe 3. For the code C 6,5 (3), a [243, 6, 162] 3 code, the construction of Proposition 3 yieds the foowing nine matrices (using Magma) giving automorphisms of C 6,5 (3) that correct eight errors, using the information set I 6,5. 5 Concusion [ 1 c P 8 = { 0 I 5 ] c (1, 1, 1, 0, 0), (2, 1, 0, 1, 0) } We incude an appendix showing the best vaues for s for an s-pd-set of size s + 1 that we have obtained for these MacDonad codes from these two constructions for some sma set of vaues for n and q and 1 u n 1. It can be seen that the second construction gives the better resut when u is greater than n 2. It shoud be emphasised that we have used a specific information set, I n,u, and that other information sets might yied better resuts. Of course the set from Proposition 1 is independent of the information set but is rather arge. 6 Appendix In the tabes to foow: for C n,u (q), q is a prime power, n, u as before, is the ength of the code, mw is the minimum weight, t is the error correcting capabiity, gb is the Gordon-Schönheim bound for the size of the PD-set for fu error correction, s 1 = f n u (for the first construction), s 2 the computationay found vaue of s for the second construction when u 3. Thus the size of the set is s 1 + 1, s 2 + 1, respectivey.

9 6 APPENDIX 9 Tabe 1: s-pd-sets of size s + 1 for C n,u (q) q n u mw t gb s 1 s

10 6 APPENDIX 10 q n u mw t gb s 1 s

11 6 APPENDIX 11 q n u mw t gb s 1 s

12 6 APPENDIX 12 q n u mw t gb s 1 s

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