Linear Programming Bounds for Almost-Balanced Binary Codes

07/16/2021

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by   Venkatesan Guruswami, et al.

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We revisit the linear programming bounds for the size vs. distance trade-off for binary codes, focusing on the bounds for the almost-balanced case, when all pairwise distances are between d and n-d, where d is the code distance and n is the block length. We give an optimal solution to Delsarte's LP for the almost-balanced case with large distance d ≥ (n - √(n))/2 + 1, which shows that the optimal value of the LP coincides with the Grey-Rankin bound for self-complementary codes. We also show that a limitation of the asymptotic LP bound shown by Samorodnitsky, namely that it is at least the average of the first MRRW upper bound and Gilbert-Varshamov bound, continues to hold for the almost-balanced case.