Constructions of cyclic constant dimension codes

Abstract

Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional properties that can be applied efficiently in encoding and decoding algorithms. It is desirable to find cyclic constant dimension codes such that both the code sizes and the minimum distances are as large as possible. In this paper, we explore the ideas of constructing cyclic constant dimension codes proposed in [2], IEEE Trans. Inf. Theory, 2016 and [17], Des. Codes Cryptogr., 2016 to obtain further results. Consequently, new code constructions are provided and several previously known results in [2] and [17] are extended.

Abelian non-cyclic orbit codes and multishot subspace codes

“…In fact, the non-Abelian orbit codes with generating group α σ [3] have cardinality upper bounded by n q n −1 q−1 and the minimum subspace distance less than or equal to 2k−2. To the best of our knowledge, such a construction leads to the best orbit codes (It is worth mentioning that the construction like the one shown in [6] leads to cyclic codes, which are not necessarily orbit codes). If there is an n, n q n −1 q−1 , n − 2, n 2non-Abelian orbit code, then the n, q n , n − 2, n 2 -Abelian non-cyclic orbit codes proposed in this work (and [29]) are always better for q − 1 ≥ n. Indeed, if q n < n q n −1 q−1 , then q n < n q n −1…”

“…Given H G and g i ∈ G, let C H (V i ) = C H (g i V ) be a subcode of C G (V ). The distance profile associated with g ∈ G and C H (V i ) is represented by the following polynomial in the indeterminate w, Note that the polynomials F w, α 3 , C α 9 (V ) and F w, α 6 , C α 9 (V ) are obtained from the interdistance sets D S C α 9 (V ), C α 9 α 3 V and D S C α 9 (V ), C α 9 α 6 V , respectively, and (27) F w, α 3 , C α 9 (V ) = F w, α 6 , C α 9 (V ) = 7w 2 + 14w 4 + 28w 6 .…”

“…This condition is necessary to ensure nested partitions. For instance, it is not possible to partition G 2 (6,3) in nested partitions concerning the action of the subgroup G = α generated by the primitive element α of F 2 6 over G 2 (6,3), since one subset in the first level of this partition has 9 elements the spread code C α (F 2 3 ) and the remaining subsets have 63 elements.…”

“…Due to the transitive action of the corresponding algebraic structure, these orbit codes can be classified as geometrically uniform subspace codes. On the other hand, in [19] and more recently in [6], constructions of constant dimension codes based on the union of cyclic orbit codes with a prescribed minimum distance are provided. In this case, it is not possible to state that these constant dimension subspace codes are orbit codes and, consequently, geometrically uniform subspace codes.…”

Finding Cliques in Projective Space: A Method for Construction of Cyclic Grassmannian Codes

“…Recently Ben-Sasson et al [1], Otal and Özbudak [7], Niu et al [9], and Chen and Liu [2] presented new methods for constructing such codes, what includes linearized polynomials, namely subspace polynomials and Frobenius mappings. A computational method for construction of cyclic Grassmannian codes was presented in [5].…”