On linear codes with random multiplier vectors and the maximum trace dimension property

Abstract

Let CC be a linear code of length nn and dimension kk over the finite field Fqm{{\mathbb{F}}}_{{q}^{m}}. The trace code Tr(C){\rm{Tr}}\left(C) is a linear code of the same length nn over the subfield Fq{{\mathbb{F}}}_{q}. The obvious upper bound for the dimension of the trace code over Fq{{\mathbb{F}}}_{q} is mkmk. If equality holds, then we say that CC has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let Ca{C}_{{\boldsymbol{a}}} denote the code obtained from CC and a multiplier vector a∈(Fqm)n{\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n}. In this study, we give a lower bound for the probability that a random multiplier vector produces a code Ca{C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n≥m(k+h)n\ge m\left(k+h), where h≥0h\ge 0 is the Singleton defect of CC. For the extremal case n=m(h+k)n=m\left(h+k), numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

Keywords

• trace codes
• subfield subcodes
• dimension of trace codes
• random alternant codes
• weight enumerator
• singleton defect
• 14g50
• 15a03