Optimal Linear Codes Over GF(7) and GF(11) with Dimension 3

Optimal Linear Codes of Dimension 4 over GF(5)

[48] __, " On complexity of trellis structure of linear block codes, " IEEE [49] T. Klgve, " Upperbounds on codes correcting asymmebic errors, " IEEE [SO] __, " Minimum support weights of binary codes, " IEEE Trans. Inform. [53] A. Lafourcade and A. Vardy, " Asymptotically good codes have infinite trellis comwlexitv. " IEEE Duns. issue on " Codes and Finite Geometries "). [59] __, " The shift b...

Optimal Linear Codes over Z

We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring Zm. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over Z8 and Z9 of lengths up to 6. We determine the minimum distances of optimal linear codes over Z4 for lengths up to 7. Some examples of optimal codes are given.

On optimal linear codes over F5

Let nq(k, d) be the smallest integer n for which there exists an [n, k, d]q code for given q, k, d. It is known that n8(4, d) = ∑3 i=0 ⌈ d/8i ⌉ for all d ≥ 833. As a continuation of Jones et al. [Electronic J. Combinatorics 13 (2006), #R43], we determine n8(4, d) for 117 values of d with 113 ≤ d ≤ 832 and give upper and lower bounds on n8(4, d) for other d using geometric methods and some exten...

Optimal linear codes, constant-weight codes and constant-composition codes over $\Bbb F_{q}$

Optimal linear codes and constant-weight codes play very important roles in coding theory and have attached a lot of attention. In this paper, we mainly present some optimal linear codes and some optimal constant-weight codes derived from the linear codes. Firstly, we give a construction of linear codes from trace and norm functions. In some cases, its weight distribution is completely determin...

Optimal linear identifying codes

Binary Linear Codes with Optimal Scaling and Quasi-Linear Complexity

We present the first family of binary codes that attains optimal scaling and quasi-linear complexity, at least for the binary erasure channel (BEC). In other words, for any fixed δ > 0, we provide codes that ensure reliable communication at rates within ε > 0 of the Shannon capacity with block length n = O(1/ε2+δ), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n). Furth...