Double circulant codes for the Lee and Euclidean distance

1.   Introduction

Double circulant codes over finite fields have been known to be asymptotically good for the Hamming distance for a long time ^[1,5]. The proof relies on a polynomial representation of double circulant codes, along with some number theoretic conjecture (Artin primitive root conjecture ^[13]). In general, the Hamming distance is not a natural metric for measuring error correction capabilities of codes over rings ^[14], or more generally of codes over alphabets of size $> 3,$ as attributing different weights to different nonzero symbols results into finer distances. For instance, the homogeneous metric is an important metric for ring alphabets. Thus, it was proved that double circulant codes over certain Galois rings are asymptotically good for the homogeneous distance ^[15].

In the present article, we shall consider double circulant codes over the rings $\mathbb{Z}_{{{p^m}}}$ where $p$ is an odd prime, for the Lee and Euclidean distances, motivated by the following considerations.

The Lee metric is instrumental in constructing insertions/deletion codes ^[18] and the Euclidean metric is instrumental in constructing spherical codes ^[19]. In both cases, we use a projection technique due to Yaglom for spherical codes ^[11], and to ^[18] for insertion/deletion codes, that maps points within a ball in some dimension to the unit sphere in dimension one more. The Lee metric controls the Manhattan metric properties of lattices in constructed from codes over $\mathbb{Z}_{{{p^m}}}$. Similarly, the Euclidean metric over $\mathbb{Z}_{{{p^m}}}^N$ controls the standard Euclidean distance in the unit sphere of $\mathbb{R}^N.$ For general background on, respectively, codes for the Lee metric, and codes for the Euclidean distance, we refer the reader respectively to ^[2,3] and to ^[7].

We derive the relative Manhattan distance and the rate of asymptotic growth for balls in the Manhattan metric when the ambient space dimension goes to infinity and the radius is fixed. We also derive the relative square Euclidean distance and the rate of spherical asymptotic growth for balls in the Euclidean metric when the radius and size go to infinity.

By expurgated random coding techniques, we derive a lower bound on the relative Lee (resp. Euclidean) distance of double circulant codes over the said rings. The counting arguments involved in the proof rely on the polynomial representation of quasi-cyclic codes, and their analysis via the CRT for polynomials developed in ^[11,12].

An outline of the paper is as follows. In Section 2, we provide some background material on Galois rings, Euclidean and Lee balls, and the algebraic structure of double circulant codes. Section 3 studies the asymptotic behavior of double circulant codes by the expurgated random coding argument made familiar by the Varshamov-Gilbert and Shannon bounds. Three cases are considered: Lee distance (small and large alphabets) and Euclidean distance. Section 4 concludes the article.

2.   Preliminaries

2.1.   Some rings

Throughout the paper, let $p$ be an odd prime and $m$ be a positive integer. The ring $\mathbb{Z}_{p^m}$ is the ring of integers modulo $p^m$. The Galois ring $GR(p^m, p^{mn})$ of order $p^{ mn},$ and characteristic $p^m$ is the Galois extension of $\mathbb{Z}_{p^m}$ with degree $n$. It is a local ring, with maximal ideal $\langle p \rangle$. The Teichmüller set ${{\mathcal{T}}} = \{{x \in GR(p^m, p^{mn}) \mid x^{p^n} = x}\}$ of $GR(p^m, p^{mn})$ is a set of representatives of the residue field $\mathbb{F}_{p^n} = GR(p^m, p^{mn}) /{{\langle p\rangle}}$. It is known that $GR(p^m, p^{mn}) = {{\mathcal{T}}}\oplus p {{\mathcal{T}}} \oplus... \oplus p^{m-1} {{\mathcal{T}}}$ so that any element $x \in GR(p^m, p^{mn})$ can be written as a base $p$ decomposition of $GR(p^m, p^{mn})$, i.e., $x = {{\alpha}}_0+p{{\alpha}}_1+...+p^{m-1}{{\alpha}}_{m-1}$ where ${{\alpha}}_i \in {{\mathcal{T}}}$ for $0\leqslant i < m$. For motivation and background see ^[20].

2.2.   Codes over $\mathbb{Z}_{q}$

A linear code ${{\mathcal C}}$ (for simply code) over $\mathbb{Z}_{q}$ of length $N$ is a submodule of $\mathbb{Z}_{q}^N$. A code of length $2N$ is called double circulant over $\mathbb{Z}_{q}$ if its generator matrix $G$ is of the form $G = (I, A)$, where $I$ is the identity matrix of order $N$ and $A$ is a circulant matrix over $\mathbb{Z}_{q}$ of the same order.

The $\mathbb{Z}_{q}$-module $\mathbb{Z}_{q}^N$ is equipped with three natural distances ^[4] that we describe in the following three paragraphs.

2.2.1.   Hamming distance

The Hamming distance between two codewords in ${{\mathcal C}}$ is the number of places where they differ, and it is denoted by $d_H$. The Hamming weight of any codeword in ${{\mathcal C}}$ is the number of nonzero components, and is denoted by $w_H$.

2.2.2.   Lee distance

The Lee weight of an element $h \in \mathbb{Z}_{q}$ viewed as an integer $\in [0, q)$ is defined as follows:

The Lee weight of vectors $\mathbb{Z}_{q}^N$ (codewords in ${{\mathcal C}}$) is the sum of Lee weight of its components. The Lee distance $d_L$ between two integers $h_1, h_2 \in \mathbb{Z}_{q}$ is the Lee weight of $h_1-h_2$. The Lee distance $d_L$ between two vectors in $\mathbb{Z}_{q}^N$ (codewords in ${{\mathcal C}}$), is the sum of the Lee distances between their corresponding components.

In ^[2], it is shown that the Lee weight of any nonzero vector ${{\bf x}} \in \mathbb{Z}_{p^m}^N$ is bounded from above and below as follows:

2.2.3.   Euclidean distance

The Euclidean weight of any element $h$ in $\mathbb{Z}_{q}$ can be defined directly as

This notion can be extended to vectors in the obvious way, and the Euclidean distance between two vectors ${{\bf x}}$ and ${{\bf y}}$ on $\mathbb{Z}_{q}^N$ is then defined as $d_E({{\bf x}}, {{\bf y}}) = w_E({{\bf x}}-{{\bf y}}).$ The minimum Euclidean distance $d_E$ of a code ${{\mathcal C}}$ is then given by

As in ^[19], if ${q} = 2s+1$, we will define the addressing map $\phi$ from $\mathbb{Z}_{q}$ into $\mathbb{R}$ by

This map can be extended to vectors in the obvious way.

We remark that the Euclidean distance between two vectors ${{\bf x}}$ and ${{\bf y}}$ on $\mathbb{Z}_{q}^N$ satisfy the property

2.3.   Insertion/deletion codes in $\mathbb{Z}^N$

Following ^[18], we define the Manhattan distance between two vectors ${{\bf x}}, {{\bf y}} \in \mathbb{Z}^N$ as

Define an $(N, d^1, \mathcal{{N}}, r)-$ code as a set of vectors of $\mathbb{Z}^n,$ with pairwise Manhattan distance at least $d^1$, every coordinate at least $r$, and total sum of entries equal to $\mathcal{{N}}$. The addressing map $Y_r$ to the ring $\mathbb{Z}_{q}$ uses the coset representatives $r, r+1, \dots, r+{q}-1$ of $q\mathbb{Z}$ into $\mathbb{Z}$. We introduce a map $\Upsilon_{\mathcal{{N}}}$ from $\mathbb{Z}^N$ to $\mathbb{Z}^{N+1}$ defined by

Note that $\mathcal{N} = \sum_{i = 1}^{N}Y_r(x_i)$.

Proposition 1. If ${{\mathcal C}}$ is a code over $\mathbb{Z}_{q}$ of length $N$ and minimum Lee distance $d_L$ over $\mathbb{Z}_{q}$, with the addressing $Y_r$, and $\mathcal{{N}}-N ({q}+r-1) > r$, then $\Upsilon_{\mathcal{{N}}}\circ Y_r ({{\mathcal C}})$ is an $(N+1, d^1, \mathcal{{N}}, r)-$ code in $\mathbb{Z}^{N+1}$ where its Manhattan distance is bounded below by the Lee distance of ${{\mathcal C}}$.

The coordinates of each codeword are at least $r$ by Definition of $Y_r.$ The last coordinate of the $\Upsilon_{{\mathcal{{N}}}}$ image is also $\ge r$ because $\sum_{i = 1}^N x_i\le N({q}+r-1)$. The total sum of the entries of the $\Upsilon_{{\mathcal{{N}}}}$ image is ${\mathcal{{N}}}$ by Definition of $\Upsilon_{{\mathcal{{N}}}}.$ The pairwise Manhattan distance of codewords is at least $d^1,$ because the Manhattan distance is bounded below by the Lee distance. Note that the $\Upsilon_{{\mathcal{{N}}}}$ adds an extra entry to vectors which can only increase the Manhattan distance.

2.4.   Spherical codes in $\mathbb{R}^N$

In this section we use ^[7], the ball of radius $e$ in Euclidean $N$-space $\mathbb{R}^N$ is the set defined as

The sphere of radius $e$ in Euclidean $N$-space $\mathbb{R}^N$ is the set defined as

When $e = 1$, $S(N, 1)$ is called the unit sphere. A spherical code $\mathcal{X}$ is a finite subset of $S(N, 1)$. The squared minimum distance $\rho$ is the smallest squared distance between pairs of distinct codewords:

Following ^[19], we embed $\mathbb{R}^N$ into $\mathbb{R}^{N+1}$ by the Yaglom map

Note that this map sends the ball $B(N+1, e)$ of radius $e$ in $\mathbb{R}^N$ into the sphere $S(N+1, e)$ with radius $e$ in $\mathbb{R}^{N+1}$.

Proposition 2. ^[19] Let $q = 2s+1$. If $\mathcal{C}$ is a code of length $N$ and minimum Euclidean distance $d_E$ over $\mathbb{Z}_{q}$, then $\mathrm{Y} (\phi (\mathcal{C}))$ is a spherical code lying in the sphere $S(N, s \sqrt{N})$ of squared Euclidean distance $d_E$.

2.5.   Algebraic structure of double circulant codes of odd length

From now on, we assume that $n$ is an odd integer, and $gcd(n, p^m) = 1$. We can cast the factorization of $x^n-1$ into distinct basic irreducible polynomials over $\mathbb{Z}_{p^m}$ in the form

where $\mu$ nonzero unit in $\mathbb{Z}_{p^m}$, $g_i(x)$ is a self-reciprocal polynomial with degree $2e_i$ for $2\leq i \leq u$, and $h^{*}_j(x)$ is the reciprocal polynomial of $h_j(x)$ with degree $b_j$ for $1 \leq j \leq t$. By the Chinese Remainder Theorem $(CRT)$, we have

Note that all of these rings are extensions of $\mathbb{Z}_{p^m}$. Let $\mathcal{R}_n = \frac{ \mathbb{Z}_{p^m}[x]}{\langle x^n-1\rangle}$. This decomposition naturally extends to $\mathcal{R}_n^ 2$ as follows

In particular, each linear code ${{\mathcal C}}$ of length $2$ over $\mathbb{Z}_{p^m}$ can be decomposed as the -CRT sum-

So ${{\mathcal C}}$ can be viewed as a submodule of $\mathcal{R}_n^2$, with generator $\langle 1, a(x)\rangle$, where the $x$- expansion of $a(x) \in \mathbb{Z}_{p^m}[x]$ is the first row of $A$.

2.6.   Codes over $\mathbb{Z}_{p}$ and asymptotic bounds

Let ${{\mathcal C}}$ be a code of length $N$ over $\mathbb{Z}_{p}$. This code is linear if it is a $\mathbb{Z}_{p}$-vector subspace of $\mathbb{Z}_{p}^N$. The dimension of a code ${{\mathcal C}}$, denoted by $k$, is equal to its dimension as a vector space.

The three parameters of a code are written compactly as $[N, k, d_H]$, $[N, k, d_L]$ or $[N, k, d_E],$ depending on the distance considered. When we extend this notation to code ${{\mathcal C}}$ that is a submodule of $\mathbb{Z}_{q}^N$, then $k = \log_{q}(|{{\mathcal C}}|)$ where $|{{\mathcal C}}|$ is the number of codewords in ${{\mathcal C}}$. Let ${{\mathcal C}}(N)$ be a family of codes with parameters $[N, k_N]$. The rate of ${{\mathcal C}}(N)$ is defined to be $R = \limsup_{N\rightarrow \infty} \frac{k_N}{N}$. The relative minimum distance $\delta$ depends on the third parameters as follows:

Remark 3. Let $s = \frac{q-1}{2}$. If ${{\mathcal C}}(N)$ a family of codes with the following distances:

(i) Hamming distance $d_{H, N}$, then its relative Hamming distance is $\delta_H = \liminf_{N\rightarrow \infty} \frac{d_{H, N}}{N}$.

(ii) Lee distance $d_{L, N}$, then is its relative Lee distance is $\delta_L = \liminf_{N\rightarrow \infty} \frac{d_{L, N}}{s N}$.

(iii) Euclidean distance $d_{E, N}$, then its relative Euclidean distance is $\delta_E = \liminf_{N\rightarrow \infty} \frac{d_{E, N}}{s^2 N}$.

A family of codes is said to be good if $R \delta \neq 0$.

Let $V_L(N, e, q)$ (resp. $V_E(N, e, q)$) denote the size of the Lee (resp. Euclidean) sphere of radius $e$, and let us define $A_L(N, d_L, q)$ (resp. $A_E(N, d_E, q)$) to be the maximal number of codewords in a code of length $N$ over $\mathbb{Z}_{q}$ with Lee (resp. Euclidean) distance $d_L$ (resp. $d_E$).

The following Theorem is the analogue of the standard Gilbert bound of the Hamming metric in the Lee and Euclidean metric. The standard proof is omitted.

Theorem 4. There are codes in $\mathbb{Z}_{q}^N$ of Lee distance $d_L$ and of cardinality

There are codes in $\mathbb{Z}_{q}^N$ of Euclidean distance $d_E$ and of cardinality

The following three theorems give the asymptotic exponent of $V_L(N, e, q)$ (resp. $V_E(N, e, q)$) for $N$ large and $e = s \tau N$, where $0 \leq \tau \leq 1$, and with the standard definition of the rate and relative minimum distance of the family of codes ${{\mathcal C}}(N)$. Their respective corollaries are the analogues of the classical Asymptotic Gilbert-Varshamov Bound ^[9] for, respectively, the Lee distance (first small alphabets, then large alphabets) or the Euclidean distance. For simplicity'sake we concentrate on the case of $p$ odd.

2.6.1.   Lee balls for small alphabets

If we denote

where $0 \leq \tau \leq 1$.

Theorem 5. ^[3] If $q = 2s+1$, then $l(\tau, q) = 1+\log_q\alpha\beta^{s\tau}$, where $\alpha \geqslant 0, \beta \geqslant 0$ satisfy

where $0 \leqslant \tau \leqslant (q+1)/2q$. Moreover, $l(\tau, q) = 0$ if $(q+1)/2q \leqslant \tau \leqslant 1$.

Corollary 6. ^[3] If $0 < \delta_L \leq \frac{q+1}{2q}$ where $q \geq 2$, then

where $R = \liminf_{n \mapsto \infty }\frac{1}{n} \log_q A_L(n, s\delta_L n, q)$.

2.6.2.   Lee balls for large alphabets

Theorem 7. ^[19] For $q\geqslant 2r+1$, for large N and $\frac{r}{N}$ tending to $\omega$, we obtain

where

Corollary 8. ^[19] If $0 < \delta_L \leq L_q(1)$ where $q \geq {2 \delta_L} N + 1$, then

2.6.3.   Euclidean balls for small alphabets

Theorem 9. ^[8] The asymptotic exponent of $V_E(N, e, q)$ for $N$ large and $e = \lfloor \tau N\rfloor$ is

where $E(\tau, q) = \log_q(f(\mu))- \tau \log_q(\mu)$ and $\mu$ is the unique real positive solution of $z f'(z) = \tau f(z)$, and $f(z) = 1+2\sum_{i = 1}^{s} z^{i^2}$.

Corollary 10. ^[8] If $0 < \delta_E \leq 1- \frac{1}{q}$ where $q = 2s+1$, then

3.   Main results

This section is only for the special case of the factorization of $x^n-1$ over $\mathbb{Z}_{p^m}$ into exactly two basic irreducible polynomials. In particular, for $m = 1,$ the existence of such a factorization for infinitely many $n'$s will depend on Artin primitive root conjecture, which is known to hold under Generalized Riemann Hypothesis (GRH), and is proved by Hath-Brown for all but a finite number of primes ^[13]. Lifting such a factorization, for fixed $n,$ from $\mathbb{F}_p$ to $\mathbb{Z}_{p^m}$ is ensured by Hensel lifting ^[20]. Let $\mathcal{C}_a$ denotes the double circulant code generated by $\langle1, a\rangle$, where $a = a(x)$ is a polynomial in $\mathbb{Z}_{p^m}[x]$.

3.1.   Long Lee double circulant codes

Theorem 11. From the above assumptions. If $f, g \in \mathbb{Z}_{p^m}^{n}$ such that $(0, 0)\neq (f, g)$, and $(f, g)$ has Lee weight less than $n$, then the vector $(f, g)$ gives at most $\lambda$ double circulant codes over $\mathbb{Z}_{p^m}$ of length $2n$ with $\lambda = p^{(m-1)n+1}$.

By the CRT, $(f, g)\simeq (f_1, g_1)\oplus(f_2, g_2)$ where $f_1$ and $g_1 = a_1 f_1$ as elements of $\mathbb{Z}_{p^{m}}$, $f_2$ and $g_2 = a_2 f_2$ as elements of $GR(p^m, p^{m(n-1)})$. Consider $\mathcal{C}_{a_1} = \langle[1, a_1]\rangle$, and $\mathcal{C}_{a_2} = \langle[1, a_2]\rangle$. Let $(f_1, g_1)\in \mathcal{C}_{a_1}$ and $(f_2, g_2)\in \mathcal{C}_{a_2}$. Determining $a$ is equivalent to determining the pair $(a_1, a_2)$.

We distinguish the following cases:

ⅰ. If $f_1$ is a unit, then $a_1$ is uniquely determined by $g_1 = a_1f_1$.

ⅱ. If $(f_1 \neq 0)$ is not a unit, where $f_1 = p^hf'_1$ such that $f'_1$ is a unit and $1 \leq h \leq m-1$, we then obtain $g_1 = p^h g^\ast$ where $g^\ast \in \mathbb{Z}_{p^m}$, which implies $a_1 f'_1-g^\ast = p^{m-h} \alpha$ where $\alpha = \sum_{i = 0}^{h-1}\alpha_i p^i$. Each $\alpha_i$ for $0 \leq i \leq h-1$ can take at most ${p}$ values. So $a_1$ takes at most $p^h \leq p^{(m-1)}$ values.

ⅲ. If $f_1 = 0$, then $a_1$ can take at most $p^m$ values.

${\rm{iii}}-1.$ If $f_2 = 0$, then $(f, g) = (0, 0)$, a contradiction as $(f, g)$ is a nonzero vectors.

${\rm{iii}}-2.$ If $f_2$ is a unit then $a_2$ is uniquely determined by the equation $g_2 = a_2f_2$. So $a$ has $p^{m}$ values.

${\rm{iii}}-3.$ If $f_2 \neq 0$ is not a unit, the case is similar to that of $f_1$ with $\alpha$ having at most $p^{h(n-1)}$ values by the $p$-adic representation in $GR(p^m, p^{m(n-1)})$. Thus $a_2$ has at most $p^{(n-1)(m-1)}$ since $h\leq m-1$. It follows that $a$ has $p^{m}p^{(m-1)(n-1)}$ values.

ⅳ. If $f_1\neq 0$ and $f_2\neq 0$ are not units, then by $(\text{ii})$ and $(\text{iii-3})$ $a$ has at most $p^{n(m-1)}$ values.

ⅴ. If $f_1\neq 0$ and $f_2\neq 0$ are units, then $a = \frac{g}{f}$ has a unique solution.

ⅵ. If $f_1\neq 0$ is not a unit and $f_2 = 0$, we then obtain $w_H(f) = n$, and by (1.4.1) we have $w_L((f, g)) > n$, a contradiction with the hypothesis.

Thus, we obtain that there are at most $\lambda = p^mp^{(m-1)(n-1)} = p^{n(m-1)+1}$ double circulant codes over $\mathbb{Z}_{p^m}$ of length $2n$.

The following results use that fact that the volumes of Lee balls for (large and small alphabets) with radius $s\delta_0 N$ are, up to subexponential terms, $p^{Nm(L_{{p^m}}(s.\delta_0) + o(1))}$ (resp. $p^{Nm(1-l(\delta_0, {{p^m}}))}$), when $0 < \delta_0 < 1$ and $N$ go to infinity.

Theorem 12. Let $p = 2s+1$ and $p^m \geqslant 2a+1$ with $a = 2sn\delta_0$ such that $\delta_0 \in(0, 1)$. Then the family of double circulant codes over $\mathbb{Z}_{p^m}$ with length $2n$, rate $\frac{1}{2}$, and relative distance $\delta_L$, satisfies $\delta_L \geq L^{-1}_{{p^m}}(\frac{1}{2m})s^{-1}$. In particular, that family of codes is good.

Let ${{p^m}} = 2s+1$ be fixed, and let $\Omega_n$ denote the size of the family of double circulant codes over $\mathbb{Z}_{p^m}$ of length $2n$. Thus, for $n\rightarrow \infty$ we have $\Omega_n \sim p^{mn}$. Assume we can prove that for $n$ large enough $\Omega_n > \lambda_n V(2n, d_{L, n}, {{p^m}})$. Here $\lambda_n = p^{n(m-1)+1}$. This would imply, by Theorem 11, that there are double circulant codes of length $2n$ in the family with minimum Lee distance greater than or equal $d_{L, n}$. Let $\delta_L$ be the relative Lee distance of this family. If we take $d_{L, n}$ to be the largest number satisfying $\Omega_n > \lambda_n V(2n, d_{L, n}, {{p^m}})$, and assume a growth of the form $d_{L, n}\sim 2 s n \delta_o$, then $\Omega_n \sim \lambda_n V(2n, d_{L, n}, {{p^m}})$ for $n\rightarrow \infty$ using the large alphabets entropic estimate for $V_L(2n, d_{L, n}, {{p^m}}) \sim p^{2nm(L_p(s\delta_o)+o(1))}$, with values of $\Omega_n$ and $\lambda_n$, with the estimate ${\delta_o} = L^{-1}_{{p^m}}(\frac{1}{2m})s^{-1}$. The result follows by observing that, by definition of $\delta_L$, $\delta_L \geq \delta_o$.

The next Theorem follows by using a similar argument as in the proof of Theorem 12.

Theorem 13. Let ${{p^m}} = 2s+1$. Then the family of double circulant codes over $\mathbb{Z}_{p^m}$ of length $2n$, and rate $\frac{1}{2}$, of relative distance $\delta_L$, satisfies $\delta_L \geq l^{-1}(\frac{2m-1}{2m}, {{p^m}})$. In particular, that family of codes is good.

By ^[8], assume that $d^1, r$ are fixed, $\mathcal{{N}} \rightarrow \infty$, and $n \sim \eta \mathcal{{N}} /r$, for some constant $\eta \in (0, 1)$. Denote by $R^1$ the asymptotic rate of a family of $(2n, d^1, \mathcal{{N}}, r)$-codes ${{\mathcal C}}_n$, and the rate is given as follows

Corollary 14. There is a family of $(2n+1, d^1, {{N}}, r)$-codes over $\mathbb{Z}$ of relative Manhattan distance at least $l^{-1}(\frac{2m-1}{2m}, {{p^m}})$, and rate $R^1$, which satisfies $R^1 = \frac{\eta m}{r} \log_2p$.

Let $d^1, r$ be fixed, by Theorem 13, there is a family of double circulant codes ${{\mathcal C}}$ denoted by $\Omega_n$ over $\mathbb{Z}_{p^m}$ of length $2n$ and of Lee distance at least $d_L$ and by Proposition 1, there is a family of $(2n+1, d^1, \mathcal{N}, r)$- codes up to normalization $\Upsilon_{\mathcal{N}}(Y_r({{\mathcal C}}))$ of $\mathbb{Z}^{2n+1}$, and its rate satisfies $R^1 = \frac{1}{\mathcal{{N}}} \log_2p^{mn} \Rightarrow R^1 = \frac{\eta m}{r} \log_2p$.

In Table 1, values of $\delta_L$ are listed for some values of ${{p^m}}$. If ${{\mathcal C}}(n)$ is a family of codes of parameters $[n, k_n, d_{L, n}]$, then by Corollary 14 $l(\delta_L, {{p^m}}) = \frac{2m-1}{2m}$.

3.2.   Long Euclidean double circulant codes

Theorem 15. If $f, g \in \mathbb{Z}_{p^m}^n$ such that $(f, g)\neq (0, 0)$, and $(f, g)$ has Euclidean weight less than $n$, then there are at most $p^{(m-1)n+1}$ polynomials $a$ such that $(f, g)\in \mathcal{C}_a$.

We can prove this result in the same manner as Theorem 11. By using Eqs (2.1) and (2.2), we get $w_H \leqslant w_L \leqslant w_E$. It follows that there are at most $p^{(m-1)n+1}$ polynomials $a$ such that $(f, g) \in {{\mathcal C}}_a$.

Theorem 16. Let ${{p^m}} = 2s+1$. Then the family of double circulant codes over $\mathbb{Z}_{p^m}$ of length $2n$, rate $\frac{1}{2}$, and relative distance $\delta_E$, satisfies $\delta_E \geq E^{-1}(\frac{1}{2m}, p)\cdot s^{-2}$. In particular, these families of codes are good.

By applying Theorem 9 in the case where the radius of the Euclidean sphere is $2s\delta_E n$, and using a similar technique to the proof of Theorem 12 the result follows.

In Table 2, values of $\delta_E$ are listed for some values of ${{p^m}}$. If ${{\mathcal C}}(n)$ is a family of Euclidean codes of parameters $[n, k_n, d_{E, n}]$, then by Theorem 16 $E(s^2 \cdot \delta_E, {{p^m}}) = \frac{1}{2m}$.

Following ^[6], let $R$ denotes the asymptotic binary rate of a family of spherical codes $\mathcal{X}$, and the rate is given as follows

In ^[10] a lower bound of Shannon is given a lower bound on the binary rate of a spherical code $R_{S}$ of a given $0 \leq \tau \leq 1$

From Theorem 16 and Proposition 2, the following corollary immediately follows.

Corollary 17. Let ${{p^m}} = 2s+1$. There is a family of spherical codes up to normalization $Y(\phi({{\mathcal C}}))$ of $\mathbb{R}^{2n+1}$, binary rate $\frac{{m}\log_2 p}{2}$, and relative squared distance $\delta_E \geq {E^{-1}(\frac{1}{2m}, p) s^{-2}}$.

In Table 3 is shown that comparison between $R(\delta_E)$ and $R_S(\delta_E)$ where $\delta_E$ is given in previous corollary.

4.   Conclusions

In this paper we have proved the existence of asymptotically good families of double circulant codes for the Lee distance and the Euclidean distance. The motivation was to construct spherical codes from Euclidean distance codes and insertions/deletion codes in the case of the Lee distance. There are certainly some more applications in the domain of Euclidean lattices, or lattices for the Manhattan metric obtained from codes by the so-called Construction A ^[11]. At a more technical level, considering quasi-cyclic codes of index higher than 2, extending these results to three- and four-circulant codes and to their negacirculant counterparts following the paths in ^[16,17] is a worthwhile project.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-Msc-28-130-40). The authrs, therefore, acknowledge with thanks DSR techical and financial support.

Conflict of interest

Prof. Patrick Solé is the Guest Editor of special issue "Mathematical Coding Theory and its Applications" for AIMS Mathematics. Prof. Patrick Solé was not involved in the editorial review and the decision to publish this article.

All authors declare no conflicts of interest in this paper.