Blocking sets, minimal codes and trifferent codes

Abstract

We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set.Over the finite field F 3 , we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length n has size at most 3 n/4.55 , improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length n and size at least 1 3 (9/5) n/4 , thus (asymptotically) matching the best lower bound on trifferent codes.We also give explicit constructions of affine blocking sets with respect to codimension-2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to F 3 , we obtain linear trifferent codes of size at least 3 7n/240 , improving the current best explicit construction that has size 3 n/112 .

Geometric dual and sum‐rank minimal codes

“…This last is in general slightly weaker than the known lower bound on the length of minimal codes (see [5,Theorem 2.14]), recently improved in [12,Theorem 1.4] and [47, Theorem A] for large k. Note that for k = 2, the above bound is sharp.…”

“…□ Remark 5.6. By Proposition 5.5, and more precisely from (12), a partition of q PG(1, ) m in a scattered linear set gives a one-weight code. Indeed, the construction of doubly extended linearized Reed-Solomon code with parameters ∕ m m [( , …, , 1, 1), 2] q q m can be read via an associated system U U ( , …, ) q 1 + 1 with the following properties:…”

“…We delve into an exploration of their parameter properties. The geometry of minimal codes have been important to construct and give bounds in both Hamming and rank metric (see [48,12,5,2,24,3]), via the so-called strong blocking sets. These, introduced first in [21] in relation to saturating sets, are sets of points in the projective space such that the intersection with every hyperplane spans the hyperplane.…”

“…The main purpose of this paper is to further study the structure, parameters and constructions of minimal codes in the sum-rank metric, recently introduced in [46]. Minimal codes are classical objects in the Hamming-metric rich in connections with different areas of mathematics, such as cryptography [38], finite geometry [1,48], and combinatorics [12]. One of the main concerns about these objects is to find bounds on their parameters.…”

“…In particular, one difficult problem is to know how short they can be and to construct short minimal codes. In [12,5,47] lower bounds on the length of minimal codes are proved, whereas in [12,2,24] the best-known upper bounds on the length of the shortest minimal codes are presented. These last are implicit existence results.…”