Codes, Cryptology and Curves with Computer Algebra

Partition-balanced families of codes and asymptotic enumeration in coding theory

“…In [31], the authors give a definition of a balanced family of codes that is a special case of Definition 2.2. In particular, it is defined with respect to the partition of X into two classes, namely, P 1 = {0} and P 2 = X\{0}.…”

Locally recoverable J-affine variety codes

“…We can extend to all r t 's the bound for r 1 given in Proposition 1.1. For it we use generalized Hamming weights (see [15,Section 4.5.1] for the definition and properties of these weights). Let us see that result.…”

The Poincaré Polynomial of a Linear Code

“…The Poincaré polynomial of a code and the multivariate Tutte polynomial of the matroid given by the arrangement attached to the code determine each other. For this result we need some results from [18,20] first.…”

“…Two projective systems on the projective line are equivalent if and only if the corresponding points are mapped to each other by a fractional transformation. Moreover if three distinct points of the system remain fixed under such a transformation, then the remaining points remain also fixed by [11] and Propositions 5.1.33 and 5.1.34 of [20]. Now suppose that the codes C(a) and C(b) are equivalent with a, b ∈ F q and a, b ∈ {0, 1}, then the projective systems P(a) and P(b) are equivalent.…”

“…It is a polynomial in several variables and gives not only the weight of a codeword but the entries contributing to that weight. We will introduce this polynomial through arrangements of hyperplanes attached to the code; we will prove that the weight enumerator can be obtained from the Poincaré polynomial (Theorem 6) and this polynomial is equivalent to the multivariate Tutte polynomial [18,20,21] of the matroid given by the above arrangement (Theorem 10).…”