FpRS-Additive Cyclic Codes are Asymptotically Good

In coding theory, the rate and the relative minimum distance are two important invariants to assess a family of codes&#x2019; asymptotic characteristics. The relative minimum distance of codes is used to measure error-correcting capacity, while the rate of codes is used to quantify the ratio of a family of codes&#x2019; information coordinates to all available coordinates. For example, a family of binary linear codes with parameters <inline-formula> <tex-math notation="LaTeX">$[n, k, d] = [{2^{r}-1, 2^{r}-1-r, 3}]$ </tex-math></inline-formula> is known as the Hamming codes, where <inline-formula> <tex-math notation="LaTeX">$r \geq 2$ </tex-math></inline-formula>. When <inline-formula> <tex-math notation="LaTeX">$n \rightarrow \infty $ </tex-math></inline-formula> then <inline-formula> <tex-math notation="LaTeX">$r \rightarrow \infty $ </tex-math></inline-formula>, we have, the rate is <inline-formula> <tex-math notation="LaTeX">$ \lim \limits _{r \rightarrow \infty } \frac {k}{n}= \lim \limits _{r \rightarrow \infty } \frac {2^{r}-1-r}{2^{r}-1}= 1$ </tex-math></inline-formula>, the relative minimum distance is <inline-formula> <tex-math notation="LaTeX">$ \lim \limits _{r \rightarrow \infty } \frac {d}{n}=\lim \limits _{r \rightarrow \infty } \frac {3}{2^{r}-1}=0$ </tex-math></inline-formula>. We have that the codes in this family have asymptotically good information rates but their relative minimum distances tend to zero, implying that they have asymptotically bad error-correcting capabilities. In general, determining the rate and relative minimum distance for a class of linear codes is not an easy task. However, with specific structures, it is possible to obtain the rate and relative minimum distance of particular families of linear codes. Because cyclic codes have a pleasant algebraic structure, people attempt to discover classes of asymptotically good cyclic codes, or demonstrate that all cyclic codes in certain classes are asymptotically bad. In this work, we look into the relative minimum distance and rate of a certain family of linear codes with specified structures. For triple alphabets, Aydogdu and Gursoy (2019), and Wu et al. (2018) investigated <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2} \mathbb {Z}_{2} \mathbb {Z}_{4} $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2} \mathbb {Z}_{4} \mathbb {Z}_{8} $ </tex-math></inline-formula>-additive cyclics codes, and Dinh et al. (2021) studied <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2}\mathbb {F}_{2}[u^{2}]\mathbb {F}_{2}[u^{3}] $ </tex-math></inline-formula>-additive cyclic codes and their applications in constructing optimal codes. Codes over triple alphabets have been examined in various works. However, the asymptotic properties of these codes have not yet been studied. Motivated by these, in this paper, we construct a class of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p}RS$ </tex-math></inline-formula>-additive cyclic codes generated by 3-tuples of polynomials, where <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula> is the finite field, <inline-formula> <tex-math notation="LaTeX">$R=\mathbb {F}_{p}+u\mathbb {F}_{p}+u^{2}\mathbb {F}_{p}+\cdots + u^{r-1}\mathbb {F}_{p}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$u^{r}=0$ </tex-math></inline-formula>) and <inline-formula> <tex-math notation="LaTeX">$S=\mathbb {F}_{p}+u\mathbb {F}_{p}+u^{2}\mathbb {F}_{p}+\cdots + u^{s-1}\mathbb {F}_{p}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$u^{s}=0$ </tex-math></inline-formula>) where p is a prime number and <inline-formula> <tex-math notation="LaTeX">$r\lt s$ </tex-math></inline-formula>. We provide their algebraic structure and show that generator matrices can be obtained for all codes of this class. Using a random Bernoulli variable, we investigate the asymptotic properties in this class of codes. Furthermore, let <inline-formula> <tex-math notation="LaTeX">$0 \lt \delta \lt 1$ </tex-math></inline-formula> be a real number and <inline-formula> <tex-math notation="LaTeX">$k, l$ </tex-math></inline-formula> and t be pairwise co-prime positive integers such that the entropy at <inline-formula> <tex-math notation="LaTeX">$\frac {(k+l+t)\delta }{3}$ </tex-math></inline-formula> is less than <inline-formula> <tex-math notation="LaTeX">$\frac {2}{3}$ </tex-math></inline-formula>, we prove that the relative minimum homogeneous distances converge to <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>, and the rates of the random codes converge to <inline-formula> <tex-math notation="LaTeX">$\frac {1}{k+l+t}$ </tex-math></inline-formula>. Consequently, <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p}RS $ </tex-math></inline-formula>-additive cyclic codes are asymptotically good.

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