TY - GEN
T1 - Cyclic subspace codes and sidon spaces
AU - Raviv, Netanel
AU - Tamo, Itzhak
N1 - Publisher Copyright: © 2017 IEEE.
PY - 2017/8/9
Y1 - 2017/8/9
N2 - The interest in subspace codes has increased in recent years due to their application in error correction for random network coding. In order to study their properties and find good constructions, the notion of cyclic subspace codes was introduced by using the extension field structure of the ambient space. However, to this date there exists no general construction with a polynomial relation between k, the dimension of the codewords, and n, the dimension of the entire space. Independently of the study of cyclic subspace codes, sSidon spaces were recently introduced by Bachoc et al. as a tool for the study of certain multiplicative properties of subspaces over finite fields. In this paper it is shown that Sidon spaces are necessary and sufficient for obtaining a full-orbit cyclic subspace code with minimum distance 2 k - 2. By presenting several constructions of Sidon spaces, full-orbit cyclic subspace codes are obtained, in which n is quadratic in k. The constructions are based on a variety of tools; namely, Sidon sets, that are sets of integers in which all pairwise sums are distinct, irreducible polynomials, and linearized polynomials. Further, the existence of a Sidon space in which n is linear in k is shown, alongside the fact that any Sidon space induces a Sidon set.
AB - The interest in subspace codes has increased in recent years due to their application in error correction for random network coding. In order to study their properties and find good constructions, the notion of cyclic subspace codes was introduced by using the extension field structure of the ambient space. However, to this date there exists no general construction with a polynomial relation between k, the dimension of the codewords, and n, the dimension of the entire space. Independently of the study of cyclic subspace codes, sSidon spaces were recently introduced by Bachoc et al. as a tool for the study of certain multiplicative properties of subspaces over finite fields. In this paper it is shown that Sidon spaces are necessary and sufficient for obtaining a full-orbit cyclic subspace code with minimum distance 2 k - 2. By presenting several constructions of Sidon spaces, full-orbit cyclic subspace codes are obtained, in which n is quadratic in k. The constructions are based on a variety of tools; namely, Sidon sets, that are sets of integers in which all pairwise sums are distinct, irreducible polynomials, and linearized polynomials. Further, the existence of a Sidon space in which n is linear in k is shown, alongside the fact that any Sidon space induces a Sidon set.
KW - Cyclic subspace codes
KW - Linearized polynomials
KW - Network Coding
KW - Sidon sets
KW - Sidon spaces
UR - http://www.scopus.com/inward/record.url?scp=85034039773&partnerID=8YFLogxK
U2 - https://doi.org/10.1109/ISIT.2017.8006635
DO - https://doi.org/10.1109/ISIT.2017.8006635
M3 - Conference contribution
T3 - IEEE International Symposium on Information Theory - Proceedings
SP - 784
EP - 788
BT - 2017 IEEE International Symposium on Information Theory, ISIT 2017
T2 - 2017 IEEE International Symposium on Information Theory, ISIT 2017
Y2 - 25 June 2017 through 30 June 2017
ER -