On Linear Codes with Random Multiplier Vectors and the Maximum Trace Dimension Property
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Let C be a linear code of length n and dimension k over the finite field 𝔽_q^m. The trace code Tr(C) is a linear code of the same length n over the subfield 𝔽_q. The obvious upper bound for the dimension of the trace code over 𝔽_q is mk. If equality holds, then we say that C 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 C_𝐚 denote the code obtained from C and a multiplier vector 𝐚∈ (𝔽_q^m)^n. In this paper, we give a lower bound for the probability that a random multiplier vector produces a code C_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), where h≥ 0 is the Singleton defect of C. For the extremal case n=m(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.
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