Some New Optimal Ternary Linear Codes

Designs, Codes and Cryptography, 12, 5 11 (1997) c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Some New Optimal Ternary Linear Codes ILIYA BOUKLIEV* Institute of Mathematics, Bulgarian Academy of Sciences, P.O.Box 323, 5000 V. Tarnovo, Bulgaria Communicated by: H. van Tilborg Received February 23, 1996; Revised August 1, 1996; Accepted September 9, 1996 Abstract. Let d 3 (n, k) be the maximum possible minimum Hamming distance of a ternary [n, k, d; 3]-code for given values of n and k. It is proved that d 3 (44, 6) = 27, d 3 (76, 6) = 48, d 3 (94, 6) = 60, d 3 (124, 6) = 81, d 3 (130, 6) = 84, d 3 (134, 6) = 87, d 3 (138, 6) = 90, d 3 (148, 6) = 96, d 3 (152, 6) = 99, d 3 (156, 6) = 102, d 3 (164, 6) = 108, d 3 (170, 6) = 111, d 3 (179, 6) = 117, d 3 (188, 6) = 123, d 3 (206, 6) = 135, d 3 (211, 6) = 138, d 3 (224, 6) = 147, d 3 (228, 6) = 150, d 3 (236, 6) = 156, d 3 (31, 7) = 17 and d 3 (33, 7) = 18. These results are obtained by a descent method for designing good linear codes. Keywords: ternary codes, optimal codes, construction method 1. Introduction Let Fq n be the n-dimensional vector space over the Galois field F q. The Hamming distance between two vectors of Fq n is defined to be the number of coordinates in which they differ. A q-ary linear [n, k, d; q]-code is a k-dimensional linear subspace of Fq n with minimum distance d. A central problem in coding theory is to optimize one of the parameters n, k, d for given values of the other two. Two versions are: Problem 1. Find d q (n, k), the largest value of d for which there exists an [n, k, d; q]-code. Problem 2. Find n q (k, d), the smallest value of n for which there exists an [n, k, d; q]-code. A code which achieves one of these two values is called optimal. Problem 2 for the case of ternary (q = 3) codes has been considered by Hill and Newton [26]. The problem of finding n 3 (k, d) was solved for k 4 for all d, and values of n 3 (5, d) were determined for all but 30 values of d. The works of van Eupen [6], [4], van Eupen and Hill [8], van Eupen, Hamada and Watamori [7], Hamada and Watamori [21], Hamada, Helleseth and Ytrehus [20], Bogdanova and Boukliev [1], and Landgev [14] solve the problem of finding n 3 (k, d) for k = 5 for all d. The problem of finding n 3 (6, d) has been studied by many authors. Results have been obtained by Brouwer and van Eupen [2], Daskalov [3], van Eupen [6], [5], Gulliver [10], * This work was partially supported by the Bulgarian National Science Fund under Grant No. MM - 502/1995. This work was presented in part at the International Workshop OCRT 95, Sozopol, Bulgaria, May 26 June 1, 1995 and at the IEEE International Simposium on Information Theory, Whistler, Canada, September 17 22, 1995.

6 BOUKLIEV Gulliver and Bhargava [11], [12], Kschischang and Pasupathy [13], Hamada [15], [16], Hamada and Helleseth [17], [18], [19], Hamada and Watamori [22], Hill [24] and Hill and Greenough [25]. This paper considers ternary codes. In Section 2 a method for construction of good linear codes is presented. Section 3 includes the new results. 2. Construction Method Let S k,q be the set of all column vectors a = (a 1, a 2,...,a k ) t Fq k such that either a 1 = 1 or a 1 = a 2 = =a i 1 = 0, a i = 1 for some integer i in {2, 3,...,k}, where k 3 and Fq k denotes a k-dimensional vector space over F q. Then S k,q consists of all (q k 1)/(q 1) nonzero vectors in Fq k and the vectors in S k,q can be regarded as (q k 1)/(q 1) points in a finite projective geometry PG(k 1,q). A linear code is called projective if no two columns of a generator matrix are linearly dependent. We regard the columns of the generator matrix of a projective code as a subset of S k,q.for any n-element subset of S k,q there is a projective [n, k 0, d; q]-code, k 0 k with a generator matrix containing these elements as columns. Denote by T (C) the set of the columns of the generator matrix of the code C, considered as elements of S k,q. The Griesmer bound [9], [28] provides an important lower bound on n q (k, d): k 1 n q (k, d) g q (k, d) = d/q i. i=0 LEMMA 1 ([23]) Suppose that d q k 1 andthatcisan[n,k,d;q]-code which attains the Griesmer bound. Then C is projective. The problem of the construction of a [g q (k, d), k, d; q] code (if such a code exists) can be formulated in the following way: Find n elements of S k,q such that the [n, k, d; q] code meets the Griesmer bound. We use the following notations: 1. Each n-element subset of S k,q is called a solution. 2. The substitution of a given element of a solution s by another one is called an elementary transformation. 3. The neighbourhood N(s) of a solution s is the set of all solutions s that can be obtained from s by an elementary transformation. 4. We define the evaluation function by f (s) = d min (s)q k + q k A dmin (s), where d min (s) is the minimum distance of the corresponding code and A dmin (s) is the number of codewords of minimum weight in the corresponding code.

SOME NEW OPTIMAL TERNARY LINEAR CODES 7 The problem of the construction of a [g q (k, d), k, d; q] code (if such one exists) if considered as a combinatorial optimization problem is the following max{ f (s), s S k,q, s =n}. The construction method is based on the next heuristic algorithm: variables s, s : solution; Number of Descents:integer; begin Number of Descents:=Random(1000); while Number of Descents>0 do begin {descent } Get Initial Solution (s); while exists s in N(s) such that f(s ) > f(s) do s:=s ; if d min (s) = d target then Output Solution (s); Number of Descents:=Number of Descents 1; end; {descent } end. The initial solution s can be chosen in several ways: (i) By using a code C with parameters [n l, k, d δ; q] where δ l. In this case l elements from S k,q, not belonging to T (C ) are randomly added to T (C ). (ii) By using a code C with parameters [n + l, k, d + l 1; q]. In this case l elements of T (C ) are randomly deleted. This algorithm can be used for constructing optimal codes not meeting the Griesmer bound. If d > q k 1 the set S k,q must be taken several times. For all codes constructed in the paper by this method the set S k,q is taken twice. Some examples are given in the next table: Initial Solution Result [120, 6, 78; 3] [124, 6, 81; 3] [124, 6, 81; 3] [130, 6, 84; 3] [160, 6, 105; 3] [156, 6, 102; 3] [156, 6, 102; 3] [152, 6, 99; 3] Other versions of the descent method are used in [11], [27]. 3. The New Results The results in dimension 6 are summarized in Theorem 1.

8 BOUKLIEV THEOREM 1 There exist codes with parameters (i) [44, 6, 27; 3], [76, 6, 48; 3], [94, 6, 60; 3], [124, 6, 81; 3], [130, 6, 84; 3], [134, 6, 87; 3], [138, 6, 90; 3], [148, 6, 96; 3], [152, 6, 99; 3], [156, 6, 102; 3], [164, 6, 108; 3], [170, 6, 111; 3], [179, 6, 117; 3], [188, 6, 123; 3],[206, 6, 135; 3] [211, 6, 138; 3], [224, 6, 147; 3], and [236, 6, 156; 3]; (ii) [228, 6, 150; 3]. Proof. (i) Codes with such parameters are constructed by the method from Section 2. Theirweight enumerators are: [44, 6, 27; 3] 1 + 352z 27 + 264z 30 + 24z 33 + 88z 36, [76, 6, 48; 3] 1 + 408z 48 + 144z 51 + 8z 54 + 168z 57, [94, 6, 60; 3] 1 + 456z 60 + 76z 63 + 192z 69 + 4z 72, [124, 6, 81; 3] 1 + 608z 81 + 96z 90 + 24z 99, [130, 6, 84; 3] 1 + 392z 84 + 216z 87 + 8z 90 + 54z 93 + 34z 96 + 22z 102 + 2z 105, [134, 6, 87; 3] 1 + 456z 87 + 160z 90 + 60z 96 + 28z 99 + 24z 105, [138, 6, 90; 3] 1 + 572z 90 + 144z 99 + 8z 108 + 4z 117, [148, 6, 96; 3] 1 + 414z 96 + 186z 99 + 54z 105 + 56z 108 + 18z 114, [152, 6, 99; 3] 1 + 430z 99 + 198z 102 + 18z 105 + 42z 108 + 40z 117, [156, 6, 102; 3] 1 + 496z 102 + 144z 105 + 8z 108 + 40z 111 + 40z 120, [164, 6, 108; 3] 1 + 648z 108 + 40z 117 + 40z 126, [170, 6, 111; 3] 1 + 418z 111 + 226z 114 + 8z 117 + 20z 120 + 20z 123 + 30z 129 + 6z 132, [179, 6, 117; 3] 1 + 424z 117 + 212z 120 + 16z 123 + 24z 126 + 16z 129 + 28z 135 + 6z 138 + 2z 141, [188, 6, 123; 3] 1 + 504z 123 + 72z 126 + 120z 132 + 8z 135 + 24z 141, [206, 6, 135; 3] 1 + 540z 135 + 172z 144 + 16z 153, [211, 6, 138; 3] 1 + 352z 138 + 198z 141 + 154z 144 + 2z 156 + 22z 162, [224, 6, 147; 3] 1 + 336z 147 + 312z 150 + 56z 153 + 24z 171 [236, 6, 156; 3] 1 + 552z 156 + 144z 159 + 8z 162 + 24z 183. The description of the corresponding generator matrices is given in Appendix. (ii) This code has been constructed as a quasi-twisted one with a generator matrix consisting of 38 twistulant matrices with first rows 010000, 110000, 210000, 101000, 201000, 111000, 210100, 120100, 220100, 101100, 201100, 111100, 211100, 202100, 112100, 122100, 222100, 111010, 211010, 121010, 221010, 212010, 122010, 210110, 120110, 121110, 221110, 112110, 212110, 122110, 111210, 121210, 221210, 112210, 222210, 111111, 121111, 112111. The weight enumerator of the code is 1 + 384z 150 + 296z 153 + 24z 156 + 24z 174. For quasi-twisted codes see [25]. All codes described above are optimal with respect to minimum distance (See [22]). THEOREM 2 d 3 (31, 7) = 17 and d 3 (33, 7) = 18. Proof. From [22] we have d 3 (31, 7) 17 and d 3 (33, 7) 18.

SOME NEW OPTIMAL TERNARY LINEAR CODES 9 Codes with parameters [31, 7, 17; 3] and [33, 7, 18; 3] were constructed by the method from Section 2. Their weight enumerators and generator matrices are [31, 7, 17; 3] 1+324z 17 +294z 18 +162z 19 +324z 20 +216z 21 +162z 22 +324z 23 +216z 24 +162z 25 +2z 27, 0000000000000011111111111111111 0000000011111100000011111122222 0000011100112200112200012200122 0001100202021200111211200200012 0110202211220100002111210200000 1022211102220002120100110000002 2221102000100010120112100201002 [33, 7, 18; 3] 1+348z 18 +162z 19 +162z 20 +378z 21 +162z 22 +162z 23 +378z 24 +162z 25 +162z 26 +110z 27, 000000000000001111111111111111110 000000001111110000001111112222220 000001110011220011220001220001221 000110020202120011121120020010120 011020221122010000211121020020000 102221110222000212010011000000021 222110200010001012011210020110020 Appendix. Generator Matrices of Codes from Theorem 1 Let the vectors from S k,q be arranged in lexicographic order. A generator matrix of a projective code is represented by a binary vector of length (q k 1)/(q 1) with 1 indicating the presence, and 0 the absence of a vector of S k,q. This binary vector is then broken into blocks of length 4, each of which is represented by a hexadecimal symbol from {0, 1,...,9,a,b,c,d,e, f}. If it is necessary the last block is beforehand completed by zeroes. Example. if q = 3 and k = 3 the set S k,q looks like that: 0000111111111 0111000111222 1012012012012 Then the code [8,3,4;3] with generator matrix: 00011111 01111122 10101212

10 BOUKLIEV is represented in the following way: 1110000111011 (2) = e1d8 (16). [44, 6, 27; 3] c0040100040000a000410000208822000010002400840842004000802042028800a8204004000020 0142010a008; [76, 6, 48; 3] 1420a12101484540028588233000421806050c80c3080200328259420bc340240001101201400402 00400c00100; [94, 6, 60; 3] 00900070480000802129d200842d2420828010415234101090600b84c451e930494710093100e310 03a10354608; [124, 6, 81; 3] 1660b9295264c374020185039c28342d1089dec0132d20c0004078000906f213c3260376c025e011 0d002133a19; [129, 6, 83; 3] 1662b9295264c274020195039c29342d1589dec0132d20c00040780c0906f213c32603768025e011 0d002133a19; [134, 6, 87; 3] 1661b9295264c37402019d039c29342d1589dec0932d20c00040780c0906f213c3262376c025e011 0d002133a19; [138, 6, 90; 3] 010241e044040c0131bdea210f0dc0804636109bead20051c8cac7e1736bfd4c47018a03108be512 946aa551005; [148, 6, 96; 3] dead645c41e21901a06a2be29d29216b511039191ae100b1a39a24a98dc560450b34a1914075a8a5 710ad281951; [152, 6, 99; 3] b5b2c052d9d2760498782b06672d02da52722b95b05114e8a8036223d369144b26a281f80b253649 034b2222b94; [156, 6, 102; 3] b5b2c0d2d9d2760498782b06672d02da52722b95b05114e8a8036223d369144b26a281f82b253649 074b2222b95; [164, 6, 108; 3] b5b2c0d2d9d27614987c2b06672d02da52722b95b05134e8e8136223d369144b26a283f82b253649 474b2223b95; [170, 6, 111; 3] 0258105c356044b3608e7ea401fdc68fc220b81aec773b12899945b646f90976b1f70bcd16826efb ad2248d2bd8; [178, 6, 116; 3] 02dfe85c357684b364ae7ea401f9c68ec220a018ec773b12899945b646f90976b1f709cd16826efb ad2268d2bdc; [188, 6, 123; 3] 23c00946aaa542cf23c2917cf5671ea541c19e721db71faf34805eee30b5db22c73add2d373acb4d d6bbad7cc47; [206, 6, 135; 3] 35f8e77d3d485b5bde09ffab016fe33ed74d9c3c761a2f13caffd5024f437df4dfe5141f96d26fbb 54f65f18121; [210, 6, 137; 3] 29d1265f7c4335e369467dae2dedf1ae49c7b9f8b1ee8e1b36d36c96b5861beefb60f6dde597e7f4 d1b6bfae12d; [224, 6, 147; 3] fba476f5c3cd0be3bced71eeead37a785ea734de5ed9ca37e7adacedd7ca3f6b6add57b1ddd71cee f8728fad3d3; [12, 6, 6; 3] the extended ternary Golay code, 00000000020000000000000000011000000100000010000000000020000000004000000080000000 800200a0000; The generator matrices of the [130,6,84;3], [179,6,117;3] and [211,6,138;3] codes are obtained by adding the vectors (0,1,0,1,2,2) t, (0,1,0,0,0,1) t and (1,2,1,2,1,2) t respectively to the generator matrices of the [129,6,83;3], [178,6,116;3] and [210,6,137;3] codes. We obtain a generator matrix of a non-projective [236,6,156;3] code concatenating the generator matrices of the [12,6,6;3] and [224,6,147;3] codes given above. Acknowledgments The author thanks Prof. S. Dodunekov, Prof. S. Kapralov and Dr. R. Daskalov for their helpful remarks.

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