Ancestor Lifting of Quasi-Cyclic Additive Codes and Certified Quantum Codes

Abstract

Additive codes over $\Ffour$ that are self-orthogonal under the trace inner product correspond to binary quantum stabilizer codes. We study 1-generator $(1,2)$-quasi-cyclic (QC) additive codes and their exact quantum distances. For composite lengths $n=\ell m$, an ancestor-lifting theorem transports dual words of the folded ancestor to the length-$n$ code. A two-layer decomposition for $\ell=2$ and a CRT splitting for odd $\ell$ yield dimension laws and explain equality, escape, and mixed-branch behavior in the certified examples. A palindromic criterion ensures symplectic self-orthogonality, while reversal pairing and a symplectic Gram rank formula describe broader construction families. Exact distances are established by information-set decoding screening, Brouwer--Zimmermann computation on the additive dual, and independent witness verification. We certify twenty-one record-breaking binary quantum codes across twenty entries of Grassl's code table, with twenty-eight certified members in total. Propagation improves thirty-four further entries, and explicit constructions are supplied for three table-tying GV entries marked Code Missing Construction. Six additional candidates are reported only with rigorous upper bounds and heuristic screening evidence. The results connect divisor-lattice structure in additive quasi-cyclic codes with reproducible certification of exact quantum distances.

The paper studies how divisor structure constrains distances of binary stabilizer codes constructed from one-generator quasi-cyclic additive codes over $\mathbb F_4$, and how to certify improvements to finite-length code tables. For $n=\ell m$, an ancestor-lifting theorem maps trace-symplectic dual words of the length-$m$ folded code into the length-$n$ dual, multiplying symplectic weight by $\ell$. This gives quantum-distance upper bounds when lifted words lie outside the stabilizer. For $\ell=2$, a two-layer decomposition introduces a linear compatibility obstruction and expresses distance through affine-coset minimization; for odd $\ell$, CRT gives independent layers. These structures distinguish distances attained by periodic lifts from those attained by nonperiodic words. Additional results include a sufficient palindromic self-orthogonality criterion, dual decompositions, and a circulant Gram-rank formula. The computational results comprise 21 certified record-breaking codes across 20 Grassl-table entries, with 28 directly certified members overall. Nineteen entries improve by one and $[[60,13,12]]_2$ improves its entry by two. Examples include $[[73,10,15]]_2$, $[[91,15,17]]_2$, and $[[93,16,17]]_2$. Standard propagation gives 34 further reported improvements and explicit constructions for three table-tying entries previously lacking constructions. Certification combines heuristic information-set-decoding screening, exact Brouwer–Zimmermann minimum-weight computation on the additive dual, and independently checked logical witnesses; the longest primary computation takes approximately $2.0\times10^6$ CPU seconds. Six additional candidates lack certified lower bounds. General lift-saturation and defect-spectrum claims remain conjectural, and the distance formulas still require minimization. Verification artifacts are available on request.