Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6

Abstract

We determine the maximum size A 2 (8, 6; 4) of a binary subspace code of packet length v = 8, minimum subspace distance d = 6, and constant dimension k = 4 to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in PG(7, 2) mutually intersecting in at most a point is 257. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size A 2 (8, 6) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well.

Johnson Type Bounds for Mixed Dimension Subspace Codes

“…Nevertheless parametric improved upper bounds for A q (v, v − 4) and A q (8, 3) have been obtained. We illustrate our results with a small (13,9) 34058 34591 34306 34056 Lemma 4 32514 Here "improved cdc" refers to Lemma 1, "ILP E/V" to the ILP of Etzion and Vardy, see Section 2, "SDP" to results based on semidefinite programming, see [2], "johnson" to the results obtained in this paper, and "bklb" to the currently best known lower bound, see [12]. If the entry of "ILP E/V" is written in italics, then the value for subspace distance d − 1 is taken.…”

Binary subspace codes in small ambient spaces

“…We continue with subspace distance d = 6. [24, Theorem 3.3(i)] implies A 2 (8, 6) = A 2 (8, 6; 4), where the latter was determined to A 2 (8, 6; 4) = 257 in [18]. For the constant-dimension case all two isomorphism types were classified in [18,Theorem 1].…”

“…[24, Theorem 3.3(i)] implies A 2 (8, 6) = A 2 (8, 6; 4), where the latter was determined to A 2 (8, 6; 4) = 257 in [18]. For the constant-dimension case all two isomorphism types were classified in [18,Theorem 1]. An essential building block is the lifted Gabidulin code G 8,4,3 , which corresponds to the unique 4 × 4 MRD code with rank distance 3 over F 2 , see [18,Theorem 10].…”

“…For the constant-dimension case all two isomorphism types were classified in [18,Theorem 1]. An essential building block is the lifted Gabidulin code G 8,4,3 , which corresponds to the unique 4 × 4 MRD code with rank distance 3 over F 2 , see [18,Theorem 10]. It can be extended by a solid in exactly two non-isomorphic ways.…”

“…Again by orthogonality, we assume δ 2 + δ 3 ≥ δ 5 + δ 6 . The case δ 2 + δ 3 = 0 was treated in [18,Theorem 1]. We first consider the cases where δ 3 is large.…”