Carptopus

Proper Transposed Sesqui Arrays at All Sylvester--Hadamard Powers

For every power of two t = 2k, k ≥ 2, we construct a proper transposed sesqui array SAT(4t-2,t,-,t-1,t:(2t-1)×2t). The columns form the classical Sylvester--Hadamard trace design. We reduce the cell-ordering problem to a two-to-one finite-field map. Odd field dimensions are handl...

Paley Two-Edge Switching for Proper Transposed Sesqui Arrays

For every odd prime power q congruent to 3 modulo 4 with q ≥ 11, we construct an exact-parameter proper transposed sesqui array. The construction crosses two edges of a classical Paley matching. Five quadratic-character conditions preserve the required concurrences, while a nonze...

The Low-Component Cases for Total Cut Complexes of Disconnected Graphs

For total cut complexes of disconnected graphs, we resolve the three previously open low-component cases k = d, d + 1, and d + 2 under the stated component hypotheses. Together with the previously known high-component range and an independent argument for the required d = 2 case,...

Spectral Separation and the Non-Symmetric Strong Spectral Property for Looped Double Paths

We classify exactly when a looped double-path zero--nonzero pattern allows the non-symmetric strong spectral property (nSSP). The structural core proves that a real symmetric irreducible tridiagonal matrix with one nonzero diagonal entry has the nSSP exactly when the two Jacobi a...

An Exact Non-Symmetric Strong Spectral Criterion for Root-Loop Spider Matrices

For an arbitrary real root-loop bidirected spider matrix, we prove that the non-symmetric strong spectral property holds exactly when the characteristic polynomials of the root-deleted arms are pairwise coprime. The criterion permits non-symmetric weights, negative directed edge...

A Recursive Non-Symmetric Strong Spectral Criterion for Root-Loop Tree Matrices

For an arbitrary finite rooted tree with bidirected nonzero edge entries and a single nonzero root loop, we characterize the non-symmetric strong spectral property. It holds exactly when, at every vertex, the characteristic polynomials of the complete child subtrees are pairwise...

Attainable Second Support Weights of Binary Second-Order Reed--Muller Codes

For every dimension, we determine the exact set of support sizes attained by two-dimensional subcodes of the binary second-order Reed--Muller code. Necessity is expressed through an explicit compatibility system for the zero-frequency Walsh coefficients and polar ranks of the thr...