Use of matroid theory to construct a class of good binary linear codes

It is still an open challenge in coding theory how to design a systematic linear (n, k) - code C over GF(2) with maximal minimum distance d. In this study, based on matroid theory (MT), a limited class of good systematic binary linear codes (n, k, d) is constructed, where n = 2^k-1 + · · · + 2^k-δ and d = 2^k-2 + · · · + 2^k-δ-1 for k ≥ 4, 1 ≤ δ <; k. These codes are well known as special cases of codes constructed by Solomon and Stiffler (SS) back in 1960s. Furthermore, a new shortening method is presented. By shortening the optimal codes, we can design new kinds of good systematic binary linear codes with parameters n = 2^k-1 + · · · + 2^k-δ - 3u and d = 2^k-2 + · · · + 2^k-δ-1 - 2u for 2 ≤ u ≤ 4, 2 ≤ δ <; k. The advantage of MT over the original SS construction is that it has an advantage in yielding generator matrix on systematic form. In addition, the dual code C^⊥ with relative high rate and optimal minimum distance can be obtained easily in this study.