Twenty exact minimum distances for generalized ℤₚ toric codes on twisted tori

Abstract

In arXiv:2602.20158, Liang and Chen build qudit CSS low-density parity-check codes from a pair of weight-three Laurent polynomials over a twisted two-dimensional torus, and tabulate sixty-five of them at p ∈ {3,5,7,11}. The distance column of those tables is the output of a randomized information-set algorithm, and its caption states that the printed values are upper bounds on the exact distance which the authors "believe are tight in practice". We prove that twenty of them are exact: for each of the twenty codes of Table [tab:cells] the minimum weight of a logical operator is exactly the printed d, including [[48,4,9]]₃, [[40,4,9]]₅, [[32,4,8]]₇ and [[40,4,10]]₁₁. All twenty agree with the printed number, so the numbers themselves are not new; what is new is the lower bound, which no source establishes. Each proof is two-sided: an explicit weight-d logical operator, certified not to be a product of stabilizers by a single pairing, together with a search over every support of size less than d in each of the two directions, of which there are 2 011 668 932 in all. The soundness of that search rests on a constructive echelon substitution that replaces the usual rank–nullity argument by an explicit coefficient vector. Every statement below is machine-checked in Lean 4.

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