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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 <tex>$d$</tex> and <tex>$n-d$</tex>, where <tex>$d$</tex> is the code distance and <tex>$n$</tex> is the block length. We give an optimal solution to Delsarte&#x0027;s LP for the almost-balanced case with large distance <tex>$d\geq(n-\sqrt{n})/2+1$</tex>, 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.

Linear Programming Bounds for Almost-Balanced Binary Codes