Optimal linear codes of dimension 4 over F

Discrete Mathematics, 1994

A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d, over a field of ~7 elements, We consider here the problem for quaternary codes (q=4), solving the problem for k< 3 for all values of d, and for k=4 for all but ten values of d.

Applied Algebra, Algebraic Algorithms and Error- …, 1997

Let nq(k, d) denote the smallest value of n for which there exists a linear [n, k, d]-code over the Galois field GF (q). An [n, k, d]-code whose length is equal to nq(k, d) is called optimal. In this paper we present some matrix generators for the family of optimal [n, 3, d] codes over GF (7) and GF (11). Most of our given codes in GF (7) are nonisomorphic with the codes presented before. Our given codes in GF (11) are all new.

IEEE Transactions on Information Theory, 2000

IEEE Transactions on Information Theory, 1996

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 drl ( 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 [18]. B. Upper Bounds Res (C, 43) = [6,4,3; 41 Res (C, 45) = [4,4,2; 41 By [4], [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 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.

Designs, Codes and Cryptography, 2007

We determine the minimum length n q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n q (k, d) = g q (k, d) + 1 for q k−1 − 2q k−1 2 −q + 1 ≤ d ≤ q k−1 − 2q k−1 2 when k is odd, for q k−1 − q k 2 − q k 2 −1 − q + 1 ≤ d ≤ q k−1 − q k 2 − q k 2 −1 when k is even, and for 2q k−1 − 2q k−2 − q 2 − q + 1 ≤ d ≤ 2q k−1 − 2q k−2 − q 2 .

Designs, Codes and Cryptography, 2010

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on q (r, 3). General constructions are given and upper bounds on q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.

Discrete Mathematics, 2004

Let [n; k; d] q -codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). In this paper, 32 new codes over GF(5) are constructed and the nonexistence of 51 codes is proved.

Finite Fields and Their Applications, 2004

We generalize a recent idea for constructing codes over a finite field F q by evaluating a certain collection of polynomials over F q at elements of an extension field. We show that many codes with the best parameters presently known can be obtained by this construction. In particular, a new linear code, a ½40; 23; 10-code over F 5 is discovered. Moreover, several families of optimal and near-optimal codes can also be obtained by this method. We call a code near-optimal if its minimum distance is within 1 of the known upper bound.

IEEE Transactions on Information Theory, 2000

Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n; n 0m; 4] -code when either 1) q = 16; m 5; m odd, or 2) q = 2 ; 7 i 15; m 4, or 3) q = 2 ;` 8; m 5; m odd. As quasi-perfect [n; n0m; 4] -codes and complete n-caps in projective spaces P G(m 01;q) are equivalent objects, new upper bounds on the size of the smallest complete cap in P G(m 01;q) are obtained.

q code be a linear code of length n, dimension k and Hamming minimum distance d over GF(q). In this paper record-breaking codes with parameters [30, 10, 15]5, [