### Optimal quaternary linear codes of dimension five

I. Boukliev, R. Daskalov, S. Kapralov
1996 IEEE Transactions on Information Theory
Let d, ( n k ) be the maximum possible minimum Hamming It is proved that d4 (33,5) = 22, d4(49 5 ) = 34, &(I31 5) = 96, d4(142,5) = 104, rla(147,5) = 108, &(I52 5 ) = 112, &(I58 5 ) = 116,d4(176,5) 2 129,d4(180,5) 2 132,&(190 5 ) 2 140,&(19j 5) = 144,d4(200,5) = 148,d4(205 5) = 132,d4(216 3 ) = 160,d4(22i 2) = = 180, and d4(247,5) = 184. A survey of the results of recent work on bounds for quaternary linear codes in dimensions four and five is made and a table with lower and upper bounds for
more » ... ( n , 5) is presented. distance of a q-ary [ rt k , d] -code for given values of n and k 168, dq(232 5) = 172, d4(237,5) 176, d4(240 3 ) = 178, d4(242 3) Index Terms-Minimum distance bounds, quaternary linear codes. c-concatenation sh-shortened code r-nonexistence of an [ n~ k ; d ; 41-code via its residual code d-nonexistence of an [ n , k , d ; 41-code follows from the nonexistence of its dual code For all the others lower bounds ( 1 5 n 5 128 ) see . B. Upper Bounds Res (C, 43) = [6,4,3; 41 Res (C, 45) = [4,4,2; 41 By , [lo] 34 5 &(49,5) 5 35. Theorem 12: d4(49,5) = 34. Proof Suppose there exists a [g4(5,35) = 49,5,35; 41-code C. codes. BY Table I nom2 of these codes exist and so By Corollary 5.1, Bl = B2 = B3 = 0. By Lemma 3 A s , = A38 = -441 = A12 = A" = A g g = 0. ( 7 1 , k ) for linear codes over field of order 4" (preliminary version)