 Hermitian self-orthogonal codes

Thu
2
Dec
Thursday 2 December, 2021 at 14:15 - 15:15
Zoom

Let C be a [n,k]_{q^2} linear code, i.e. a k-dimensional subspace of the n-dimensional vector space over the finite field with q^2 elements F_{q^2}. The code C is linearly equivalent to a Hermitian self-orthogonal code if and only if there are non-zero a_i in the finite field with q elements F_{q} such that a_1 u_1 (v_1)^q+…+a_n u_n (v_n)^q=0 for all u and v in C. For any linear code C of length n over the finite field with q^2 elements, Rains defined the puncture code P(C) to be

P(C)={a=(a_1,…,a_n) in (F_q)^n : a_1 u_1 (v_1)^q+…+a_n u_n (v_n)^q=0 for all u and v in C }.

There is a truncation of a linear code C over F_{q^2} of length n to a linear over F_{q^2} of length r <= n which is linearly equivalent to a Hermitian self-orthogonal code if and only if there is an element of P(C) of weight r. Rains was motivated to look for Hermitian self-orthogonal codes, since there is a simple way to construct a [[ n,n-2k]]_q quantum code, given a Hermitian self-orthogonal code. This construction is due to Ketkar et al, generalising the F_4-construction of Calderbank et al.

In this talk, I will detail an effective way to calculate the puncture code. I will outline how to prove various results about when a linear code has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code and how to extend it to one that does in the case that it has no such truncation. In the case that the code is a Reed-Solomon code, it turns out that the existence of such a truncation of length r is equivalent to the existence of a polynomial g(X) in F_{q^2}[X] of degree at most (q-k)q-1 with the property that g(X)+g(X)^q has q^2-r distinct zeros in F_{q^2}.

(Joint work with Ricard Vilar)