A rank-metric code C ⊆ 𝔽_(qᵐ)ⁿ is rank-metric intersecting when the rank supports of any two nonzero codewords meet nontrivially (Bartoli, Borello, Marino and Scotti, arXiv:2507.00569); a nondegenerate [n,k,d]_(qᵐ/q) code has this property if and only if i…
Leuenberger and Albrizzio (arXiv:2606.21425) show that the weight spectrum of the Reed–Muller code RM(7,14) contains every even integer between 312 and 2¹⁴-312, together with the classical small and large weights, with the possible exception of a set M of t…
Let O be a conic of PG(2,q), q odd, and let m(q) be the largest m for which some m points off O can be added to it so that the union is a (q+1+m,3)-arc, i.e. meets every line in at most three points. Dönmez, Akpinar and Pelen (arXiv:2608.18192) ask for m(q)…
For an [n,k]_q linear code C with parity-check columns h₁,…,hₙ ∈ 𝔽_qⁿ⁻ᵏ, the t-th generalized covering radius ρₜ(C) of Elimelech, Firer and Schwartz is the least r such that any t syndromes lie together in the span of at most r of the columns. Li and Xiong…
Let boldsymbol(H) be a parity-check matrix of a binary linear code of covering radius 2. A (2,0)-partition of its column set is a partition into nonempty blocks such that every syndrome, the zero syndrome included, is a sum of at most two columns lying in d…
Let n=2m, let α be a primitive element of 𝔽_(2ⁿ), let sₜ=Tr(αᵗ) be the binary m-sequence it generates, and let C_d(τ) be the crosscorrelation between {sₜ} and its decimation by d. For the Niho decimation d=4(2ᵐ-1)+1=2ᵐ⁺²-3, Helleseth, Katz and Li proved th…
For q=3ᵐ and n=q-1, let C_((1,e)) be the ternary cyclic code of length n with generator polynomial m_α(x)m_(αᵉ)(x). Ding and Helleseth reduced the question of when C_((1,e)) attains the sphere-packing-optimal parameters [n, n-2m, 4] to two root conditions o…
A function F:𝔽₂ⁿ → 𝔽₂ᵐ is kth-order sum-free if the sum of its values over every affine k-dimensional flat of 𝔽₂ⁿ is nonzero; for k=2 this is almost perfect nonlinearity. Heering, Kaspers and Taranchuk showed that a {k-1,k}-order sum-free (n,m)-function…
