## SEMINAR:Small weight codewords of projective geometric codes

### SEMINAR:Small weight codewords of projective geometric codes Title:Small weight codewords of projective geometric codes

Date/Time: 7 April 2021 / 13:40 - 14:30

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Abstract:Consider the finite projective space PG(n, q), where q is a power of some prime p. Choose integer parameters j and k such that 0 ≤ j < k < n. We denote the set of all m-dimensional subspaces of PG(n, q) by Gm. Construct the incidence matrix M whose rows are indexed by the k-spaces Gk and whose columns are indexed by the j-spaces Gj . Write a 1 if the corresponding j-space is completely contained in the corresponding k-space and a 0 otherwise. We consider the matrix over a field with characteristic p, usually the prime field Fp. Symbolically,

M ∈ FpGk×Gj                            Mκ,λ ={1 if λ ⊂ κ, 0 otherwise .

Now consider the linear code Cj,k(n, q), generated by the rows of M. Its dual code Cj,k(n, q)

⊥ consists of all vectors v such that Mv = ~0. We are interested in the small weight codewords of these codes. The (Hamming) weight of a vector is defined as the number of positions in which the vector has a non-zero entry. For linear codes, determining the minimum weight is equivalent to determining the minimum distance of the code, one the three most important parameters. We present new results, linking the problem of determining the minimum weight and characterising minimum weight codewords to a more restricted set of parameters j, k, n [ADSW20, AD21] We also revisit some old results of the codes and try to add something new to it.

Bio: [ADSW20] S. Adriaensen, L. Denaux, L. Storme, and Zs. Weiner. Small weight code words arising from the incidence of points and hyperplanes in PG(n, q). Des. Codes Cryptogr., 88(4):771–788, 2020.

[AD21] S. Adriaensen, L. Denaux. Small weight codewords in projective geometric codes. J. Combin. Theory Ser. A, Volume 180, May 2021.