We prove that the intersection number n(h)=|S∩ (S+h)| is even for every h∈𝔽_q^* , where S is the subgroup of quartic residues in 𝔽_q^* , under the assumptions that q is an odd prime power with q≡ 516 and k=4 . This resolves Conjecture 9 of [2], where the binary codes C_q,k derived from the neighborhood designs of Paley-type bipartite graphs P(q, k) were studied for an odd prime q; our result also extends the conclusion of that conjecture from odd primes to odd prime powers.
We construct infinite families of binary or ternary linear complementary dual (LCD) codes, which arise from some families of self-orthogonal codes associated with simplicial complexes or multi-variable functions. We find explicit criteria for self-orthogonal codes to preserve self-orthogonality after augmentation. For the construction of LCD codes, we either directly use some families of self-orthogonal codes or we use the augmented codes of these self-orthogonal code families. As a result, we obtain several infinite families of binary or ternary LCD codes, which include LCD-optimal families and at least LCD-almost optimal families; we also present their weight distributions. We mention that our LCD code families are new compared with the data summarized in [27, Table 5] based on the previously known LCD code families.
We describe a new, highly optimized implementation of number theoretic transforms on processors with SIMD support (AVX, AVX-512, and Neon). For any prime modulus p and any order of the form r = 2^i · 3^j |p - 1 , our implementation can automatically generate a dedicated codelet to compute the number theoretic transform of order r over 𝔽_p . New speed-ups were achieved by relying heavily on non-normalized modular arithmetic and allowing for orders r that are not necessarily powers of two.
