Affine-Half Extensions of Partial-Spread Minimal Binary Linear Codes with Explicit Weight Distributions

Source abstract

In this paper, we derive complete weight distributions and solve constrained optimization problems for affine-half coordinate extensions of a known family of partial-spread minimal binary codes. The extended code has length 2m+2m−1−1 and dimension m+1, and it remains minimal. The repetition step and its preservation of minimality are elementary; the principal results instead consist of an affine Walsh-sum identity, exact translation incidence counts, and the resulting complete weight distributions when the affine normal lies inside or outside the union of the dual-spread components. The construction and weight distributions hold for 2≤r≤2t−1. In the principal range 2≤r≤2t−2, where m=2t, we characterize every optimal coordinate set for each prescribed extension size and solve the affine-half optimization exactly.