Linear binary codes with intersection properties

Intersection of Isomorphic Linear Codes

Given an (n, k) linear code C over GF(q), the intersection of C with a code ?(C), where ? # Sn , is an (n, k1) code, where max[0, 2k&n] k1 k. The intersection problem is to determine which integers in this range are attainable for a given code C. We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we giv...

New Binary Linear Codes

Binary Linear Block Codes

Binary linear complementary dual codes

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual [n, k] codes with the largest minimum weight among all binary linear complementary dual [n, k] codes. We characterize binary linear complementary dual codes with the largest minimum weight for small dimensions. A complete ...

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...