INTRODUCTION  The problem of classification of optimal ternary constant-weight codes (TCW) is considered  We use combinatorial and computer methods to find inequivalent codes for some cases for 3 ≤ d ≤ n ≤ 9

PRELIMINARIES  Ternary (n,M,d) code - consists of M vectors of length n over alphabet {0,1,2} such that any two codewords differ in at least d positions  (n,M,d,w) constant-weight code - all the codewords have the same Hamming weight w  A 3 (n,d,w) – the largest possible value of M when the other parameters are fixed Codes with such parameters are called optimal  One of the main open problem of algebraic coding theory is enumeration (up to equivalence) of optimal codes

HISTORY  V. Zinoviev (1984) – Generalization of Johnson bound for constant- weight code  A.Brouwer, J.Shearer, N.Sloane, and W.Smith (1990) - Table of binary constant-weight codes for n ≤ 28  E.Agrell, A.Vardy, and K.Zeger (2000) – Upper Bounds, Binary codes  G.Bogdanova (2000) – New bounds, TCW codes  P.Ostergard and M.Svanstrom (2002) - Table of TCW codes  D. Smith, L.Hughes and S.Perkins (2006) - New Table of Constant Weight Codes for n ≥ 28  Y.Chee, S.Ling (2007) - Constructions for q-ary Constant-Weight Codes

PRELIMINARIES  A q (n,d,n) = A q-1 (n,d)   Johnson bounds: −

ENUMERATION OF TCW CODES  Two ternary constant-weight codes are equivalent if one of them can be obtained from the other by transformations of the following types: – permutation of the coordinates of the code; – permutation of the alphabet symbols appearing in a fixed position  Exhaustive search is not applicable for large parameters

ENUMERATION OF TCW  Maximum Clique search problem – vertex set corresponds to the words of length n and Hamming weight w – two vertices are joined by an edge if the Hamming distance between the corresponding words is greater than or equal to d  (n,M,d,w) code can be shortened to get (n - 1, M',d,w) subcode  Construct a code C - by classifying all such subcodes, and then use the clique-finding approach to find the rest of the words in C.

ENUMERATION OF TCW  Two basic steps: – Finding all inequivalent possibilities for subcode C 0 – Extending any of them to the size of C  Аlgorithms are implemented in the computer package QPlus  Some of the results are also verified using Q-Extension software

RESULTS Theorem (a) There exist unique (up to equivalence) TCW codes with parameters: (3,3,3,2), (4,4,3,2), (4,2,4,2), (4,8,3,3), (4,2,4,3), (5,5,3,2), (5,2,4,2), (5,12,3,3), (5,5,4,3), (5,2,5,3), (5,5,4,4), (5,2,5,4), (6,3,4,2), (6,4,5,3), (6,2,6,3), (6,15,4,4), (6,4,5,4), (6,3,6,4), 6,12,4,5), (6,3,5,5), (6,2,6,5), (7,3,4,2), (7,14,4,3), (7,2,6,3), (7,2,7,4), (7,2,7,5), (7,7,5,6), (7,2,6,6), (7,2,7,6), (8,4,4,2), (8,5,5,3), (8,2,6,3), (8,2,8,4), (8,2,8,5), (8,2,8,6), (8,16,5,7), (8,2,8,7), (9,4,4,2), (9,3,6,3), (9,3,7,4), (9,2,8,4), (9,5,7,5), (9,3,8,5), (9,2,9,5), (9,3,9,6), (9,3,8,7), (9,2,9,7), (9,3,7,8),(9,2,9,8)

RESULTS Let #(n,M,d,w) denote the number of inequivalent TCW codes with the specified parameters. (b) We have: #(5,10,3,4)=64, #(6,6,3,2)=2, #(6,18,3,3)=54, #(6,8,4,3)=3, #(6; 24; 3; 5) ≥ 20, #(7,7,3,2)=2, #(7,4,5,3)=2, #(7,7,5,4)=45, #(7,3,6,4)=3, #(7,3,6,5)=4, #(7,9,5,5)=2, #(7; 14; 4; 6) ≥ 74, #(8,8,3,2)=3, #(8,5,6,4)=2, #(8,2,7,4)=2, #(8,8,6,5)=5, #(8,3,7,5)=3, #(8,8,6,6)=22, #(8,3,7,6)=2, #(8,4,6,7)=2, #(8,2,7,7)=2, #(9,9,3,2)=4, #(9,6,5,3)=2, #(9,6,7,6)=12, #(9,3,8,6)=4, #(9,5,7,7)=11, #(9,2,8,8)=2

EXAMPLE Operation done in: 0:3:21 Result: Vector space: 830 Total of 7883 constant-weight codes found with parameters: (Q = 3; N = 8; M = 8; D = 6; W = 5;) 5 non equivalent codes found.

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