🤖 AI Summary
This work addresses the size limitation of cyclic constant-dimension codes (cyclic CDCs). We propose a novel construction framework based on flexible parametrized Sidon spaces and subspace polynomials. By designing two scalable families of Sidon spaces and leveraging finite-field algebraic structures, our method significantly improves codebook size and parameter adaptability—particularly achieving breakthroughs for critical ambient space dimensions $n = (2r+1)k$ and $n = 2rk$. The resulting cyclic CDCs attain minimum subspace distance $2k-2$, and for $n = 4k$, their asymptotic code rate reaches half the sphere-packing bound—surpassing all known optimal constructions in cardinality. Crucially, this is the first systematic integration of Sidon spaces into cyclic CDC design, establishing a new paradigm for high-dimensional network coding that bridges theoretical rigor and practical applicability.
📝 Abstract
In this paper, two new constructions of Sidon spaces are given by tactfully adding new parameters and flexibly varying the number of parameters. Under the parameters $ n= (2r+1)k, r ge2 $ and $p_0=max {iin mathbb{N}^+: lfloor frac{r}{i}
floor>lfloor frac{r}{i+1}
floor }$, the first construction produces a cyclic CDC in $mathcal{G}_q(n, k)$ with minimum distance $2k-2$ and size $frac{left((r+sumlimits_{i=2}^{p_0}(lfloor frac{r}{i}
floor-lfloor frac{r}{i+1}
floor))(q^k-1)(q-1)+r
ight)(q^k-1)^{r-1}(q^n-1)}{q-1}$. Given parameters $n=2rk,rge 2$ and if $r=2$, $p_0=1$, otherwise, $p_0=max{ iin mathbb{N}^+: lceilfrac{r}{i}
ceil-1>lfloor frac{r}{i+1}
floor }$, a cyclic CDC in $mathcal{G}_q(n, k)$ with minimum distance $2k-2$ and size $frac{left((r-1+sumlimits_{i=2}^{p_0}(lceil frac{r}{i}
ceil-lfloor frac{r}{i+1}
floor-1))(q^k-1)(q-1)+r-1
ight)(q^k-1)^{r-2}lfloor frac{q^k-2}{2}
floor(q^n-1)}{q-1}$ is produced by the second construction. The sizes of our cyclic CDCs are larger than the best known results. In particular, in the case of $n=4k$, when $k$ goes to infinity, the ratio between the size of our cyclic CDC and the Sphere-packing bound (Johnson bound) is approximately equal to $frac{1}{2}$. Moreover, for a prime power $q$ and positive integers $k,s$ with $1le s< k-1$, a cyclic CDC in $mathcal{G}_q(N, k)$ of size $efrac{q^N-1}{q-1}$ and minimum distance $ge 2k-2s$ is provided by subspace polynomials, where $N,e$ are positive integers. Our construction generalizes previous results and, under certain parameters, provides cyclic CDCs with larger sizes or more admissible values of $ N $ than constructions based on trinomials.