We study upper bounds on the length of $\mathbb F_q$-linear QMDS codes in the folded Hamming distance relative to their other parameters, especially the field size $q$. Via a correspondence between such codes and families of subspaces, we relate the length problem to that of upper bounding $1$-subspace packings with respect to the other parameters, especially the field size. Our main result is a reduction from these families to partial spreads, which allows us to import sharp bounds from finite geometry, including results of Drake-Freeman, N\u{a}stase-Sissokho, and Honold-Kiermaier-Kurz. As a consequence, we recover the Griesmer-type upper bound on the length of QMDS codes by Ball et al. and obtain tighter upper bounds in several parameter regimes.
We study linear codes for insertion and deletion (insdel) errors through the lens of evaluation codes. We develop a general framework for analyzing random puncturings of evaluation codes, where the edit distance is controlled by only the size of the evaluation domain and the maximum number of zeros of a nonzero functio...
Quadratic ball discrepancy defines an energy on codes in finite Hamming spaces. At perfect-code parameters, its exact minimizers are the perfect codes. We fix the alphabet size, length, and code cardinality and compare all codes with these parameters. We prove tiling-defect stability: excess discrepancy above the perfe...
It is demonstrated that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.
It is demonstrated that curves with many rational points allow for nearly optimal codes over significantly smaller alphabets, and the existence of random AG codes that approach the half-Singleton bound with high probability over fields of size, breaking the linear field size barrier.
We provide a positive answer to a conjecture proposed by Toh\v{a}neanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classica...
New Delsarte-type linear programming bounds for $q$-ary $(r,\delta)$-locally recoverable codes (LRCs) are derived with three attributes: the variable set is comparable in size to that of the classical Delsarte LP, and the convex-hull LP yields improvements not captured by the previous LP and often sharpens the shorteni...
