2026/07/19 by Joshua Steier
Computer Science · Mathematics · #cs.IT #math.CO #math.IT
Let A:\mathbbF2n→\mathbbF2m be a binary linear map with fixed coordinate bases, let CA=ker A, and let λA(y) be the minimum Hamming weight of a preimage of the syndrome y. We define Shatq,s(A) as the least common check support of a q-dimensional syndrome subspace whose every nonzero element has coset-leader weight at least s. It therefore distinguishes release of q independent syndromes from release of a subspace with no easy linear combination. Deleting check coordinates F releases ker A_F/ker A, canonically isomorphic to (im A)[F]. Finiteness implies Rq(CA)≥ N2(q,s), where N2(q,s) is the shortest length of a binary code of dimension q and distance at least s; profile-Griesmer bounds independently control common check support. The hierarchy is coordinate-relabeling invariant but can change under a change of check basis. For the pair-repetition code Cn=\(x,x):x∈\mathbbF2n\, the standard realization H0=[In In] has Shatq,s(H0)=N2(q,s) whenever feasible. For every q≥ 1 and s≥ 2, with n=N2(q,s), a row-equivalent realization of the same code has value q. For a simplicial coboundary map A=δk, check erasure is top-face erasure and the released quotient is emergent cohomology. At s=1 the hierarchy reduces to generalized Hamming weights and is Tutte-determined; for s≥ 2, even identical labeled cut codes can have different values.