A Go-Up Code Construction from Linear Codes Yielding Additive Codes for Quantum Stabilizer Codes

Abstract

Given a code C over the finite field $\mathbb {F}_q$, where q is a power of a prime number, some constructions exist that permit us to obtain a new code from C over $\mathbb {F}_q$ or over a subfield of $\mathbb {F}_q$, such as subfield subcodes. However in some important applications, one needs codes over an extension field, for example in quantum error-correcting codes (QECC). We propose a technique that we call Go-Up construction, which allows us to obtain an additive or a linear code over $\mathbb {F}_{q^m}$ from any set of m linear codes over $\mathbb {F}_q$. We show under what condition this code is a self-orthogonal or self-dual code. Thus we are able to give new constructions of quantum stabilizer codes from our codes that are additive. We present several such classes of QECC. Our GU codes also have applications to algebraic coding theory, finite geometries, finite group theory, and also to combinatorial objects such as strongly regular graphs, and few-weight codes (see [3]).