The degree of functions and weights in linear codes

AbstractProperties of the weight distribution of low-dimensional generalized Reed–Muller codes are used to obtain restrictions on the weight distribution of linear codes over arbitrary fields. These restrictions are used in non-existence proofs for ternary linear code with parameters [74,10,44] [82,6,53] and [96,6,62]

Distance Verification for Classical and Quantum LDPC Codes

The techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes. Then, these techniques are considered for classical and quantum (stabilizer) low-density-parity-check (LDPC) codes. New complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of LDPC codes. These bounds are expressed in terms of the erasure-correcting capacity of the corresponding ensemble. We also present a new irreducible-cluster technique that can be applied to any LDPC code and takes advantage of parity-checks’ sparsity for both the classical and quantum LDPC codes. This technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances, which also include all known families of the quantum stabilizer LDPC codes