New constructions of cyclic subspace codes
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A subspace of a finite field is called a Sidon space if the product of any two of its nonzero elements is unique up to a scalar multiplier from the base field. Sidon spaces, introduced by Roth et al. (IEEE Trans Inf Theory 64(6): 4412-4422, 2018), have a close connection with optimal full-length orbit codes. In this paper, we present two constructions of Sidon spaces. The union of Sidon spaces from the first construction yields cyclic subspace codes in 𝒢_q(n,k) with minimum distance 2k-2 and size r(⌈n/2rk⌉ -1)((q^k-1)^r(q^n-1)+(q^k-1)^r-1(q^n-1)/q-1), where k|n, r≥ 2 and n≥ (2r+1)k, 𝒢_q(n,k) is the set of all k-dimensional subspaces of 𝔽_q^n. The union of Sidon spaces from the second construction gives cyclic subspace codes in 𝒢_q(n,k) with minimum distance 2k-2 and size ⌊(r-1)(q^k-2)(q^k-1)^r-1(q^n-1)/2⌋ where n= 2rk and r≥ 2. Our cyclic subspace codes have larger sizes than those in the literature, in particular, in the case of n=4k, the size of our resulting code is within a factor of 1/2+o_k(1) of the sphere-packing bound as k goes to infinity.
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