Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by $2\sqrt{q}$. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of $\mathrm{PG}(2, q)$. The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.
We give a polynomial-size linear programming bound for $q$-ary all-symbol locally recoverable codes with locality parameters $(r,\delta)$, without assuming linearity. The key idea is to keep, for every ordered pair of codewords and every selected recovery view, the joint Hamming weight on the helper set, the recovered coordinate, and the rest of the code -- rather than collapsing this triple into a single distance, as earlier formulations do. Averaging this three-block distribution over recovery views of the same length yields exact identities linking it to the global distance distribution, together with nonnegative product-Krawtchouk constraints that encode locality and spectral positivity simultaneously. The resulting LP has polynomially many variables, its optimum dominates the ordinary Delsarte bound, and an earlier outside-distance formulation, the convex-hull bound of Li--Wei--Xiong, and the dual-based bound of Gruica--Jany--Ravagnani all arise from it as coarser marginals. Exact rational certificates over $q=2,3,4$ show the bound is strictly stronger than the best of these prior LPs in thirteen of fifteen tested cases, pinning down seven exact maximum code sizes and twelve exact maximum linear dimensions.
For $q\in\set{2,3}$, we show that a $k$-dimensional linear code over the finite field $\F_q$ of order $q$ is linear complementary dual (LCD) exactly when one root-of-unity value of its weight enumerator has magnitude $q^{k/2}$. We convert the phase of this value, together with the parity type in the binary case, into exact linear constraints on the weight distribution and incorporate them into a Gauss-phase linear program. The resulting program uses only the ordinary weight distributions of the code and its dual and adds only a constant-size set of branch equations to the usual Hamming/MacWilliams constraints, so it remains close in size to the standard Hamming LP while retaining additional arithmetic information. Computations over the audited binary and ternary ranges show systematic strengthening of the Hamming LCD relaxation. In the binary case, comparison with the established mixed joint-weight-enumerator LP yields four strict improvements, lowering the benchmark upper bound by one in each case. Each strict comparison is verified exactly by rational feasibility witnesses and integer Farkas certificates.
A de Bruijn sequence is the cyclic prototype of a Cayley-graph observation problem: when does the ordered label word on a translated window $gY$ determine the vertex $g$? We distinguish three parameters. The unrestricted number $\operatorname{sep}_q(G)$ minimizes an arbitrary separating pattern; the connected number $\operatorname{csep}_q(G,S)$ requires a connected Cayley window containing $Y_S=\{1\}\cup S$; and the one-step number $\chi_1(G,S)$ fixes $Y_S$ and minimizes the alphabet. Thus $\operatorname{sep}_q$ is a group-level baseline, $\operatorname{csep}_q$ measures the cost of locality, and $\chi_1$ tests the smallest prescribed local window. The organizing theme is the tension between information and locality. Carbon tori test the gap between $\operatorname{sep}_q$ and $\operatorname{csep}_q$: for generalized dihedral groups $\mathbb{F}_{\ell^d}^{\times}\rtimes C_2$ we prove, for odd prime powers $\ell$, the sharp baseline $\operatorname{sep}_\ell=d+1$ and construct connected zig-zag windows, while the order-$14$ Heawood torus satisfies $\operatorname{sep}_4=2$ and $\operatorname{csep}_4=4$. The spherical $A_5$ example and a finite simple-group comparison test the fixed one-step window: explicit symmetric cubic generating tuples give $\chi_1(A_5,S)=3$ and $\chi_1(\operatorname{PSL}_2(\mathbb{F}_7),S)=4$, both at the counting bound, with structured matrix-coefficient certificates. Cyclic-coset packings, finite-field coordinates, and restricted matrix coefficients are used only as the construction tools these two examples require.
Ming-Hsuan Kang, Yun-Hsuan Hsieh· 0 citations
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