On self-dual and LCD quasi-twisted codes of index two over a special chain ring

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On self-dual and LCD quasi-twisted codes of index two over a special chain ring

Minjia Shi, Liqin Qian, Patrick Sole

To cite this version:

Minjia Shi, Liqin Qian, Patrick Sole. On self-dual and LCD quasi-twisted codes of index two over a

special chain ring. Cryptography and Communications - Discrete Structures, Boolean Functions and

Sequences , Springer, 2019, 11, pp.717 - 734. �10.1007/s12095-018-0322-5�. �hal-02411619�

https://doi.org/10.1007/s12095-018-0322-5

On self-dual and LCD quasi-twisted codes of index two over a special chain ring

Liqin Qian

²

· Minjia Shi

^1,2

· Patrick Sol ´e

³

Received: 26 April 2018 / Accepted: 25 July 2018 / Published online: 13 August 2018

©Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract

Let q be a prime power, and let F

q

denote the finite field of order q. Consider the chain ring R

_k

= F

q

[ u ] / u

^k

with k ≥ 1 an integer. We study self-dual and LCD quasi-twisted codes of index two and twisting constant λ over R

k

for the metric induced by the standard Gray map. Some special factorizations of x

^m

− λ over R

k

are studied. By random coding, we obtain four classes of asymptotically good self-dual λ-circulant codes and four classes of asymptotically good LCD λ-circulant codes over R

_k

.

Keywords Double circulant codes · Gray map · Self-dual codes · LCD codes · Quasi-twisted codes

Mathematics Subject Classification (2010) 94 B15 · 94 B25 · 05 E30

This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

Minjia Shi

smjwcl.good@163.com Liqin Qian

qianliqin 1108@163.com Patrick Sol´e

sole@enst.fr

1 Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, Anhui University, No.3 Feixi Road, Hefei, Anhui, 230039, China

2 School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China

3 CNRS/LAGA, University of Paris 8, 93 526 Saint-Denis, France

1 Introduction

Self-dual codes are important for a number of practical and theoretical reasons, as wit- nessed by [1, 3, 12]. Another important class of codes defined by their duality properties is that of Linear codes with Complementary Duals (LCD), which were introduced by Massey in 1992 for information theoretic reasons (see [18]). They have found applications recently in Boolean masking [6, 7]. Massey [18] shows that LCD codes are asymptot- ically good. LCD codes are universal: for q > 3 there is an algorithm that turns any linear code into an equivalent LCD code [5]. Still, it is of interest to find direct meth- ods of construction of LCD codes as in [4]. One such method is to use quasi-cyclic and quasi-twisted codes. In that direction, self-dual double circulant (resp. double negacircu- lant) codes over finite fields have been studied recently in [1, 3], from the viewpoint of enumeration and asymptotic performance. Some classes of quasi-twisted codes have been studied over finite chain rings in [19]. In [2], A. Alahmadi et al. have studied the linear complementary-dual multinegacirculant codes. Motivated by the techniques in [1–3, 8, 19, 20], we use the Chinese Remainder Theorem (CRT) approach to quasi-twisted codes as introduced in [11, 13, 14]. In particular, we study two classes of self-dual (resp. LCD) negacirculant codes of index 2 over R

k

. Combining with [19], we study two families of factorizations of x

^m

− λ over R

_k

with m an odd prime, gcd(m, q) = 1. When these spe- cial factorizations are thus enforced, we derive exact enumeration formulae, and obtain asymptotic lower bounds on the minimum Hamming distance of the Gray image of these codes.

The material is arranged as follows. In Section 2, we give some background materials on the ring R

k

and study the case when the element − 1 is a square in R

k

. In Section 3, we derive the enumeration formulae of self-dual (resp. LCD) double circulant and dou- ble negacirculant codes of co-index m and we study the special factorizations of x

^m

− λ with m an odd prime, and gcd(m, q) = 1. Then, we also obtain the enumeration formu- lae of self-dual (resp. LCD) double λ-circulant codes. In Section 4, we derive a modified Varshamov-Gilbert bound on the relative distance of the codes considered, building on exact enumeration results. Finally, Section 5 contains conclusions and open problems.

2 Preliminaries

2.1 The ring R

_k

= F

q

[ u ] / u

^k

Let q be a prime power, and let F

q

denote the finite field of order q. Consider the local ring R

_k

= F

q

[ u ] / u

^k

where u

^k

= 0 with unique maximal ideal u . In double λ-circulant codes case, we will consider the chain ring R

_k

= F

q

[ u ] / u

^k

when it contains a square root of − 1.

Theorem 2.1 If a

₀

+ ua

₁

+ · · · + u

^k−1

a

_k−1

∈ R

_k

= F

q

[ u ] / u

^k

is a square root of − 1 if and only if

(1) q is a power of 2, a

₀²

= − 1, a

₁

= a

₂

= · · · = a

k−2

2

= 0, where a

k 2

, a

k+2

2

, · · · , a

_k−1

∈ F

q

when k is even; a

²₀

= − 1, a

1

= a

2

= · · · = a

k−1

2

= 0, a

k+1 2

, a

k+2

2

, · · · , a

_k−1

∈ F

q

, when k is odd; or

(2) q = p

^κ

where p ≡ 1(mod 4) or q = p

^2κ

where p ≡ 3(mod 4), a

₀²

= − 1, a

₁

= a

₂

=

· · · = a

_k−1

= 0.

Proof Note that the condition is obviously sufficient. To prove its necessity, when q is a power of 2, we have (a

0

+ a

1

u + · · · + a

_k−1

u

^k−1

)

²

= a

₀²

+ a

₁²

u

²

+ · · · + a

²_k−1

u

^2(k−1)

=

− 1,when k is even, a

²₀

= − 1, a

₁

= a

₂

= · · · = a

k−2 2

= 0, a

k

2

, a

k+2

2

, · · · , a

_k−1

∈ F

q

; when k is odd, a

²₀

= − 1, a

₁

= a

₂

= · · · = a

k−1

2

= 0, a

k+1 2

, a

k+2

2

, · · · , a

_k−1

∈ F

q

. When q is a power of an odd prime, we have (a

₀

+ a

₁

u + · · · + a

_k−1

u

^k−1

)

²

= a

₀²

+ 2a

₀

a

₁

u + (2a

₀

a

₂

+ a

²₁

)u

²

+ · · · + (a

0

a

k−1

+ a

1

a

k−2

+ · · · + a

k−1

a

0

)u

^k−1

= − 1,where a

i

∈ F

q

, 0 ≤ i ≤ k − 1.

Then we get a

₀²

= − 1, a

1

= a

2

= · · · = a

k−1

= 0. Thus a

0

is a square root of − 1 over F

q

if and only if q ≡ 1(mod 4).

2.2 Codes

A linear code C over R

_k

of length n is an R

_k

-submodule of R

_kⁿ

. If x = (x

₁

, x

₂

, · · · , x

_n

) and y = (y

₁

, y

₂

, · · · , y

_n

) are two elements of R

_kⁿ

, their standard (Euclidean) inner product is defined by

x, y =

n i=1

x

_i

y

_i

, and their Hermitian scalar inner product is defined by

x, y

H

=

n i=1

x

i

y ¯

i

,

where the operation is performed in R

k

. For all z = z

0

+ uz

1

+ · · · + u

^k−1

z

_k−1

∈ F

q^2Q

+ u F

q^2Q

+ · · · + u

^k−1

F

q^2Q

, the conjugation of z over F

q^2Q

+ u F

q^2Q

+ · · · + u

^k−1

F

q^2Q

is z = z

^q₀^Q

+ uz

^q₁^Q

+ · · · + u

^k−1

z

^q_k−1^Q

, where Q is a positive integer. The Euclidean (resp.

Hermitian) dual code of C is denoted by C

^⊥

(resp. C

^⊥H

) and defined as C

^⊥

= { y ∈ R

_kⁿ

| x, y = 0, ∀ x ∈ C } (resp. C

^⊥H

= { y ∈ R

_kⁿ

| x, y

H

= 0, ∀ x ∈ C } ).

A linear code C of length n over R

_k

is called a self-dual code (resp. Hermitian self- dual code) if C = C

^⊥

(resp. C = C

^⊥H

). A linear code C of length n over R

_k

is called a linear code with complementary dual (LCD) if C

C

^⊥

= { 0 } or C

C

^⊥H

= { 0 } . A matrix A over R

k

is said to be λ-circulant if its rows are obtained by successive λ- shifts from the first row. In this paper, we consider double λ-circulant codes over R

_k

, that is [ 2m, m ] codes with generator matrices G = (I, A) with A an m × m λ-circulant matrix, we can view such a code as an R

_k

-module in R

_k²

, generated by (1, h) with the first row of A being the x-expansion of h in the ring

_x^Rm^k[−λ^x]

.

If C(m) is a family of codes with parameters [ m, k

m

, d

m

] over F

q

, the rate ρ and relative distance δ are defined as ρ = lim sup

m→∞

km

m

and δ = lim inf

m→∞

dm

m

, respectively. A family of codes is good if ρδ > 0.

In number theory, Artin’s conjecture on primitive roots states that a given integer q which is neither a perfect square nor − 1 is a primitive root modulo infinitely many primes [16].

This was proved conditionally under the Generalized Riemann Hypothesis (GRH) by Hoo- ley [9]. Hence, we can get infinite families of double λ-circulant codes C(m) over R

_k

where the analysis is made for x

^m

− 1 with a special factorization.

Recall the q -ary entropy function defined for 0 ≤ ˜ t ≤

^q−_q¹

by H

q

(˜ t ) =

0, if t ˜ = 0,

˜

tlog

_q

(q − 1) − ˜ t log

_q

(˜ t) − (1 − ˜ t)log

_q

(1 − ˜ t), if 0 < t ˜ ≤

^q−1_q

.

This quantity is instrumental in the estimation of the volume of high-dimensional Hamming balls when the base field is F

q

. The result we are using is that the volume of the Hamming ball of radius t m ˜ is asymptotically equivalent, up to subexponential terms, to q

^mHq(˜t)

, when 0 < t < ˜ 1, and m goes to infinity [10, Lemma 2.10.3].

2.3 Gray map

Any integer z can be written uniquely in base p as z = p

0

(z) + pp

1

(z) + p

²

p

2

(z) + · · · , where 0 ≤ p

i

(z) ≤ p − 1, i = 0, 1, 2, . . .. The Gray map : R → F

^pp^k−1

is defined as follows:

(a) = (b

₀

, b

₁

, b

₂

, . . . , b

_pk−1−1

),

where a = a

0

+ a

1

u + · · · + a

k−1

u

^k−1

. Then for all 0 ≤ i ≤ p

^k−2

− 1, 0 ≤ τ ≤ p − 1,we have

b

_ip+τ

=

⎧ ⎨

⎩

a_k−1+^k−2

l=1p_l−1(i)al+τ a0, if k≥3,

a1+τ a0, ifk=2.

Note that, more generally, Gray maps have been defined at the level of finite chain rings in [15, 23], linking codes over rings to codes over finite fields. For instance, when p = k = 2, it is easy to check that the Gray map adopted in the trace codes of [21] is the same as the Gray map defined here. As an additional example, when p = k = 3, write (a

₀

+ a

₁

u + a

₂

u

²

) = (b

₀

, b

₁

, b

₂

, · · · , b

₈

). According to the definition above, we have 0 ≤ i ≤ 2, 0 ≤ τ ≤ 2 and

k

−2 l=1

p

_l−1

(i)a

l

= p

0

(i)a

1

= ia

1

. Then we get

b

0

= a

2

, b

1

= a

2

+ a

0

, b

2

= a

2

+ 2a

0

, b

3

= a

2

+ a

1

, b

4

= a

2

+ a

1

+ a

0

, b

5

= a

2

+ a

1

+ 2a

₀

, b

6

= a

2

+ 2a

₁

, b

7

= a

2

+ 2a

₁

+ a

0

, b

8

= a

2

+ 2a

₁

+ 2a

₀

. It is easy to extend the Gray map from R

^m_k

to F

^pp^k−1m

, and we also know from [22] that is injective and linear.

3 Algebraic structure of λ-circulant codes of index two

In this section, we study the exact enumeration of the double self-dual and LCD λ-circulant codes over R

k

.

3.1 Double circulant codes (λ = 1)

In this subsection, we assume m is an odd integer and gcd(m, q) = 1. We can cast the factorization of x

^m

− 1 into distinct basic irreducible polynomials over R

_k

= F

q

[ u ] / u

^k

in the form

x

^m

− 1 = δ(x − 1)

s

i=2

g

_i

(x)

t

j=1

h

_j

(x)h

^∗_j

(x), (1)

where δ is a unit in R

_k

, the polynomial g

_i

(x) is self-reciprocal of degree 2e

_i

for 2 ≤ i ≤ s, and h

^∗_j

(x) is the reciprocal polynomial of h

j

(x) with degree d

j

for 1 ≤ j ≤ t. By the Chinese Remainder Theorem (CRT), we have

R

k

[ x ]

x

^m

− 1 R

k

[ x ] x − 1 ⊕

_s

i=2

R

k

[ x ] / g

i

(x)

⊕

⎛

⎝

^t

j=1

R

k

[ x ] / h

j

(x) ⊕ R

k

[ x ] / h

^∗_j

(x) ⎞

⎠

F

q

[ u, x ] u

^k

, x − 1 ⊕

_s

i=2

F

q

[ u, x ] u

^k

, g

_i

(x)

⊕

⎛

⎝

^t

j=1

F

q

[ u, x ]

u

^k

, h

_j

(x) ⊕ F

q

[ u, x ] u

^k

, h

^∗_j

(x)

⎞

⎠ R

_k

⊕

_s

i=2

( F

q^2ei

+ u F

q^2ei

+ · · · + u

^k−1

F

q^2ei

)

⊕

⎛

⎝

^t

j=1

F

_qdj

+ u F

_qdj

+ · · ·

+ u

^k−1

F

_qdj

⊕ ( F

_qdj

+ u F

_qdj

+ · · · + u

^k−1

F

_qdj

) ⎞

⎠ := R

k

⊕

_s

i=2

R

k(2e_i)

⊕

⎛

⎝

^t

j=1

(R

_k(dj)

⊕ R

_k(dj)

)

⎞

⎠ .

Note that all of these rings are extensions of R

_k

. This decomposition naturally extends to (

_x^Rm^k[x−1^]

)

²

as

R

k

[ x ] x

^m

− 1

2

R

k

⊕

_s

i=2

R

²_k(2e

i)

⊕

⎛

⎝

^t

j=1

R

²_k(d

j)

⊕ R

²_k(d

j)

⎞

⎠ .

In particular, each linear code C of length 2 over

^R_xm^k[x]−1

can be decomposed as the “CRT sum”

C C

₁

⊕

_s

i=2

C

_i

⊕

⎛

⎝

^t

j=1

(C

_j

⊕ C

_j

)

⎞

⎠ ,

where C

1

is a linear code over R

k

of length 2, C

i

is a linear code over R

k(2ei)

of length 2 for each 2 ≤ i ≤ s, and C

_j

and C

_j

are linear codes over R

k(d_j)

of length 2 for each 1 ≤ j ≤ t , which are called the constituents of C.

Lemma 3.1 Keep the same notations as above, then

(1) C

₁

is LCD if and only if 1 + r

²

∈ R

^×_k

with C

₁

= (1, r) ; (2) C

_i

is LCD if and only if 1 + η η ¯ ∈ R

^×_k(2e

i)

with C

_i

= (1, η) ; (3) C

_j

⊕ C

_j

are LCD if and only if 1 + η

η

∈ R

_k(d^×

j)

with C

_j

= (1, η

) and C

_j

= (1, η

) .

Proof (1) “ =⇒ ” If C

₁

is LCD, suppose 1 + r

²

∈ R

^×_k

, then 1 + r

²

∈ u . We have

u

^k−1

(1 + r

²

) = 0,i.e., u

^k−1

(1, r), (1, r) = 0,then u

^k−1

(1, r) ∈ C

^⊥₁

,which implies

u

^k−1

(1, r) ∈ C

1

∩ C

₁^⊥

, a contradiction.

“ ⇐= ” If 1 + r

²

∈ R

^×_k

, assume C

₁

is not LCD, then C

₁

∩ C

^⊥₁

= { 0 } . Hence, there exists r

∈ R

^×_k

such that r

(1, r) ∈ C

₁

∩ C

^⊥₁

, then r

(1, r), (1, r) = r

(1 + r

²

) = 0,since 1 + r

²

∈ R

^×_k

, then r

= 0, a contradiction.

(2) “ =⇒ ” If C

i

is LCD, suppose 1 + η η ¯ ∈ R

_k(2e^×

i)

, then 1 + η η ¯ ∈ u . We have u

^k−1

(1 + η η) ¯ = 0,i.e., u

^k−1

(1, η), (1, η)

H

= 0,then u

^k−1

(1, η) ∈ C

^⊥_i ^H

,which implies u

^k−1

(1, η) ∈ C

i

∩ C

_i^⊥H

, a contradiction.

“ ⇐= ” If 1 + η η ¯ ∈ R

_k(2e^×

i)

, assume C

_i

is not LCD, then C

_i

∩ C

_i^⊥H

= { 0 } . If η ∈ R

_k(2e^×

i)

,then C

_i^⊥H

= (1, −

_η¹_¯

) ,there exist k

1

, k

2

∈ R

^×_k(2e

i)

such that k

1

(1, η) = k

₂

(1, −

¹_η¯

),i.e., k

₁

(1 + η η) ¯ = 0. And 1 + η η ¯ ∈ R

^×_k(2e

i)

, then k

₁

= 0, a contradiction. If η ∈ u ,we can let η = ua

1

+ u

²

a

2

+ · · · + u

^k−1

a

_k−1

, a

i

∈ F

q^2ei

, 0 ≤ i ≤ k − 1,then the generator matrix of C

_i^⊥H

is of the form

− (u a ¯

1

+ u

²

a ¯

2

+ · · · + u

^k−1

a ¯

_k−1

) 1

0 u

^k−1

k

₃

, where k

₃

∈ F

q^2ei

. Thus, there exist k

₄

, k

₅

, k

₆

∈ R

_k(2e^×

i)

such that k

₄

(1, ua

₁

+ u

²

a

₂

+

· · · + u

^k−1

a

k−1

) = k

5

( − (u a ¯

1

+ u

²

a ¯

2

+ · · · + u

^k−1

a ¯

k−1

), 1) + k

6

(0, u

^k−1

k

3

),we then obtain

k

4

= − (u a ¯

1

+ u

²

a ¯

2

+ · · · + u

^k−1

a ¯

_k−1

)k

₅

,

(ua

1

+ u

²

a

2

+ · · · + u

^k−1

a

_k−1

)k

4

= k

5

+ u

^k−1

k

3

k

6

, (2) by (2), we get k

4

∈ R

^×_k(2e

i)

,but − (u a ¯

1

+ u

²

a ¯

2

+ · · · + u

^k−1

a ¯

_k−1

)k

5

∈ u , a contradiction.

(3) “ =⇒ ” If C

_j

⊕ C

_j

is LCD, assume 1 + η

η

∈ R

^×_k(d

j)

, then 1 + η

η

∈ u . We have u

^k−1

(1 + η

η

) = 0,i.e., u

^k−1

(1, η

), (1, η

) = 0,then u

^k−1

(1, η

) ∈ C

_j^⊥

(or u

^k−1

(1, η

) ∈ C

_j^⊥

), which implies u

^k−1

(1, η

) ∈ C

_j

∩ C

_j^⊥

(or u

^k−1

(1, η

) ∈ C

_j

∩ C

_j^⊥

), a contradiction.

“ ⇐= ” If 1 + η

η

∈ R

^×_k(d

j)

, assume C

_j

⊕ C

_j

is not LCD, then C

_j

∩ C

_j^⊥

= { 0 } ,

C

_j

∩ C

_j^⊥

= { 0 } . If C

_j

∩ C

_j^⊥

= { 0 } , then there exists k

∈ R

^×_k(d

j)

such that k

(1, η

) ∈ C

_j

∩ C

_j^⊥

,i.e., k

(1, η

), (1, η

) = k

(1 + η

η

) = 0, a contradiction.

Theorem 3.2 Let m denote a positive odd integer, and q a prime coprime with m. If x

^m

− 1 can be factored into irreducible polynomials over R

_k

as in (1), where m = 1 +

^s

i=2

2e

_i

+ 2

t j=1

d

_j

. Then

(1) the total number of self-dual double circulant codes over R

_k

is B

s

i=2

(q

^ei

+ 1)q

^ei(k−1)

t

j=1

(q

^dj

− 1)q

^dj(k−1)

, where

1) when q is a power of 2, B = 2q

^k2

, k is even, or B = 2q

^k−12

, k is odd;

2) when q is a power of odd prime, B = 2.

(2) the total number of LCD double circulant codes over R

_k

is (q − 2)q

^k−1

s

i=2

(q

^2ei

− (q

^ei

+ 1))q

^2(k−1)ei

t

j=1

(q

^2kdj

− q

^2(k−1)dj

(q

^dj

− 1)).

Proof (1) We can count the number of self-dual double circulant codes by counting their constituent codes.

Let (1, r) be the generator of the self-dual code C

1

over R

k

. By Theorem 2.1, when q is a power of 2, the number of r is equal to 2q

^k2

, where k is even (2q

^k−12

,where k is odd); when q is a power of odd prime, the number of choices for r is equal to 2.

Let (1, c

_ei

) be the generators of Hermitian self-dual codes C

_i

over R

_k(2ei)

, 2 ≤ i ≤ s, then (1, c

ei

), (1, c

ei

)

H

= 1 + c

ei

c

ei

= 0. Let c

ei

= c

0

+ uc

1

+ · · · + u

^k−1

c

_k−1

, where c ∈ F

_q²ei

, 0 ≤ ≤ k − 1, we then have

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

c

0

c

^q₀^ei

= − 1, c

₀

c

^q₁^ei

+ c

₁

c

₀^qei

= 0,

c

0

c

^q₂^ei

+ c

1

c

₁^qei

+ c

2

c

₀^qei

= 0,

c

₀

c

^q₃^ei

+ c

₁

c

₂^qei

+ c

₂

c

₁^qei

+ c

₃

c

₀^qei

= 0,

c

0

c

^q₄^ei

+ c

1

c

₃^qei

+ c

2

c

₂^qei

+ c

3

c

₁^qei

+ c

4

c

^q₀^ei

= 0, .. .

c

0

c

^q_k−1^ei

+ c

1

c

^q_k−2^ei

+ · · · + c

_k−1

c

^q₀^ei

= 0.

(3)

⇐⇒

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

N orm(c0)=−1, T r(c0c^q₁^ei)=0,

T r(c0c^q₂^ei)+N orm(c1)=0, T r(c0c^q₃^ei)+T r(c1c₂^qei)=0,

T r(c0c^q₄^ei)+T r(c1c₃^qei)+N orm(c2)=0, ...

T r(c₀c^q_k−1^ei )+T r(c₁c_k−2^qei )+· · ·+T r(ck−2 2 ck

2)=0, whenkis even, or T r(c0c^q_k−^ei₁)+T r(c1c_k^q₋^ei₂)+· · ·+T r(ck−3

2 ck+1

2 )+N orm(ck−1

2 )=0, whenkis odd,

where the N orm() and T r() are maps norm and trace from F

_q²ei

to F

q^ei

. So there are q

^ei

+ 1 roots for N orm(c

₀

) = − 1 and q

^ei

choices for c

_i

for 1 ≤ ≤ k − 1. Clearly, the number of solutions of (3) is equal to (q

^ei

+ 1)q

^ei(k−1)

.

As for reciprocal pairs, note that a pair (h

_j

(x), h

^∗_j

(x)) both of degree d

j

leads to counting dual pairs of codes (for the Euclidean inner product) of length 2 over R

k(d_j)

, that is to count the number of solutions of 1 + c

_d

j

c

_d

j

= 0, where (1, c

_d

j

) and (1, c

_d

j

) are the generators of C

_j

and C

_j

, respectively. If c

_d

j

∈ R

_k(d^×

j)

, then c

_d

j

= −

_c¹ dj

, there are | R

_k(d^×

j)

| = (q

^dj

− 1)q

^dj(k−1)

choices for (c

_d

j

, c

_d

j

). If c

_d

j

∈ R

k(d_j)

\ R

_k(d^×

j)

, then c

_d

j

= ux

1

+ u

²

x

2

+ · · · + u

^k−1

x

_k−1

∈ u . In this case, 1 + c

_d

j

c

_d

j

= 0, which is impossible.

(2) The code C

1

is an LCD code, by Lemma 3.1 (1), we can get 1 + r

²

∈ R

_k^×

. Let r =

r

₀

+ ur

₁

+· · ·+ u

^k−1

r

_k−1

∈ R

_k

,then 1 + r

²

= 1 + (r

₀

+ ur

₁

+ u

²

r

₂

+· · ·+ u

^k−1

r

_k−1

)

²

=

1 + r

₀²

+ 2r

₀

r

₁

u + (2r

₀

r

₂

+ r

₁²

)u

²

+ · · · + (r

₀

r

_k−1

+ r

₁

r

_k−2

+ · · · + r

_k−1

r

₀

)u

^k−1

∈ R

_k^×

. Hence, the number of r is equal to (q − 2)q

^k−1

.

The code C

i

is an LCD code, by Lemma 3.1 (2), we can get 1 + η η ¯ ∈ R

^×_k(2e

i)

. Let η = η

0

+ uη

1

+· · ·+ u

^k−1

η

_k−1

,then 1 + η η ¯ = 1 + (η

0

+ uη

1

+· · ·+ u

^k−1

η

_k−1

)(η

₀^qei

+ uη

^q₁^ei

+ · · · + u

^k−1

η

^q_k−1^ei

) = 1 + η

^q₀^ei+1

+ u(η

0

η

^q₁^ei

+ η

1

η

₀^qei

) + · · · + u

^k−1

(η

0

η

_k−1^qei

+ η

1

η

^q_k−2^ei

+ · · · + η

_k−1

η

^q₀^ei

) ∈ R

_k(2e^×

i)

. Hence, the number of η is equal to (q

^2ei

− (q

^ei

+ 1))q

^2(k−1)ei

.

Next, we count the number of LCD double circulant codes of length 2 over R

_k(dj)

for the pairs h

_j

(x) and h

^∗_j

(x) with deg(h

_j

(x)) = deg(h

^∗_j

(x)) = d

_j

. By Lemma 3.1 (3), we then get

C_j

C_j^⊥= {0}, C_j

C_j^⊥= {0}.

⇐⇒ 1 + η

η

∈ R

_k(d^×

j)

. Without loss of generality, we discuss on the unit character of η

as follows.

1) If η

∈ R

^×_k(d

j)

, then η

∈ −

_η¹

+ R

^×_k(d

j)

and |

⁻_η¹

+ R

^×_k(d

j)

| = | R

_k(d^×

j)

| = q

^(k−1)dj

(q

^dj

− 1).

Hence, there are | R

^×_k(d

j)

|

²

= q

^2(k−1)dj

(q

^dj

− 1)

²

choices for (η

, η

).

2) If η

∈ R

k(dj)

\{ R

^×_k(d

j)

∪{ 0 }} , let η

= η

₀

+ uη

₁

+· · ·+ u

^k−1

η

_k−1

, then η

= uη

₁

+ u

²

η

₂

+

· · ·+ u

^k−1

η

_k−1

, where η

1

can’t be all zero, 1 ≤

1

≤ k − 1, η

2

∈ F

_qdj

, 0 ≤

2

≤ k − 1.

We then have 1 + η

η

= 1 + uη

₁

η

₀

+ u

²

(η

₁

η

₁

+ η

₂

η

₀

) +· · ·+ u

^k−1

(η

₁

η

_k−2

+ η

₂

η

_k−3

+

· · · + η

_k−1

η

₀

) ∈ R

_k(d^×

j)

. Thus, there are (q

^(k−1)dj

− 1)q

^kdj

choices for (η

, η

).

3) If η

= 0, then η

∈ R

_k(dj)

, thus there are q

^kdj

choices for η

.

Hence, the number of the last case about reciprocal pairs is q

^2(k−1)dj

(q

^dj

− 1)

²

+ (q

^(k−1)dj

− 1)q

^kdj

+ q

^kdj

= q

^2kdj

− q

^2(k−1)dj

(q

^dj

− 1). The proof of the theorem is now completed.

3.2 Double negacirculant codes (λ = − 1)

In this subsection, assume m is an even integer and gcd(m, q) = 1, where q is a prime power. We can cast the factorization of x

^m

+ 1 into distinct basic irreducible polynomials over R

_k

as follows.

x

^m

+ 1 =

s

i=1

g

_i

(x)

t

j=1

h

_j

(x)h

^∗_j

(x), (4)

where ∈ R

_k^×

, g

_i

(x) = g

^∗_i

(x) with deg(g

_i

(x)) = 2e

_i

, 1 ≤ i ≤ s, and h

^∗_j

(x) is the reciprocal polynomial of h

_j

(x) with deg(h

_j

(x)) = deg(h

^∗_j

(x)) = d

_j

, 1 ≤ j ≤ t. Using the same notations and argument as in Subsection 3.1, we can easily carry out the result as follows:

R

_k

[ x ] x

^m

+ 1

_s

i=1

R

_k(2ei)

⊕

⎛

⎝

^t

j=1

(R

_k(dj)

⊕ R

_k(dj)

)

⎞

⎠ ,

and

C

_s

i=1

C

_i

⊕

⎛

⎝

^t

j=1

(C

_j

⊕ C

_j

)

⎞

⎠ .

Theorem 3.3 Let m denote a positive even integer, and q a prime power coprime with m.

The factorization of x

^m

+ 1 over R

_k

is of the form (4) with m =

^s

i=1

2e

_i

+ 2

t

j=1

d

_j

. Then (1) the total number of self-dual double negacirculant codes over R

_k

is

s

i=1

(q

^ei

+ 1)q

^ei(k−1)

t

j=1

(q

^dj

− 1)q

^dj(k−1)

. (2) the total number of LCD double negacirculant codes over R

k

is

s

i=1

(q

^2ei

− (q

^ei

+ 1))q

^2(k−1)ei

t

j=1

(q

^2kdj

− q

^2(k−1)dj

(q

^dj

− 1)).

Proof This proof is similar to that of Theorem 3.2, so we omitted it here.

Now, we consider a special factorization of x

^m

+ 1, where m is a power of 2, q is an odd prime. According to [17, Theorem 1] and [2, Theorems 5.1,5.3], we know that x

^m

+ 1 can be factored into two (resp. four) basic irreducible polynomials, which are reciprocal of each other over R

_k

, by limiting the size of and U , because F

q

is a subring of R

_k

. We can get the following lemma.

Lemma 3.4 Let m be a power of 2, q ≡ ± 1(mod 4).

(1) If q = 2

²

e ± 1, e is odd, then x

^m

+ 1 factors into two basic irreducible polynomials over R

k

as follows.

x

^m

+ 1 = h(x)h

^∗

(x)

with deg(h(x)) = deg(h

^∗

(x)) =

^m₂

. In this case, the number of self-dual (resp. LCD) double negacirculant codes over R

k

is

(q

^m2

− 1)q

^m(k−1)2

(resp.q

^km

− q

^m(k−1)

(q

^m2

− 1))).

(2) If q = 2

³

e ± 1, e is odd, then x

^m

+ 1 factors into four basic irreducible polynomials over R

_k

as follows.

x

^m

+ 1 = h

1

(x)h

^∗₁

(x)h

2

(x)h

^∗₂

(x) (5) with deg(h

1

(x)) = deg(h

^∗₁

(x)) = deg(h

2

(x)) = deg(h

^∗₂

(x)) =

^m₄

. In this case, the number of self-dual (resp. LCD) double negacirculant codes over R

_k

is

(q

^m4

− 1)

²

q

^m(k−1)2

(resp.(q

^km2

− q

^m(k−1)2

(q

^m4

− 1))

²

).

3.3 Quasi-twisted codes of index two (λ = 1 + ωu

^t

)

In this subsection, we focus on the case (1 + ωu

^t

) = (1 + ωu

^t

)

⁻¹

= (1 − ωu

^t

). According to [19], x

^m

− (1 + ωu

^t

) can be uniquely expressed as

x

^m

− (1 + ωu

^t

) = ςg

1

(x)

s

i=2

g

i

(x)

t

j=1

h

j

(x)h

^∗_j

(x), (6)

where m is an odd, then g

₁

(x) = x − (1 + ωu

^t

), ς ∈ R

^×_k

, g

_i

(x) = g

^∗_i

(x) with

deg(g

_i

(x)) = 2e

_i

, 2 ≤ i ≤ s, and h

^∗_j

(x) is the reciprocal polynomial of h

_j

(x) with

deg(h

j

(x)) = deg(h

^∗_j

(x)) = d

j

, 1 ≤ j ≤ t.

In fact, we notice that a (1 + ωu

^t

)-QT code over R

_k

is self-dual only if 1 + ωu

^t

= 1 − ωu

^t

, i.e., 2ωu

^t

= 0 =⇒ char(R

k

) = 2 over R

k

.

Conjecture 3.5 Assume that m is an odd prime and gcd(m, q) = 1, where q is a prime power. Let α | (m − 1) and ord

_m

(q) =

^m−1_α

, we can cast the factorization of x

^m

− λ into distinct basic irreducible polynomials over R

_k

=

^F_u^q[u]k

as follows.

(1) If α is an odd integer, then we have x

^m

− λ = A(x)

α

i=1

g

i

(x),where g

i

(x) = g

_i^∗

(x), deg(g

i

(x)) =

^m_α⁻¹

;

(2) If α is an even integer, then we have x

^m

− λ = A(x)

α

2

j=1

h

_j

(x)h

^∗_j

(x),where deg(h

_j

(x)) = deg(h

^∗_j

(x)) =

^m−1_α

; if

(i) λ = 1, A(x) = x − 1, or (ii) λ = − 1, A(x) = x + 1, or

(iii) λ = 1 + ωu

^t

, q is a power of 2, A(x) = x + 1 + ωu

^t

, where t ≥

^k₂

, ω ∈ R

^×_k

. Now, we only give some examples to illustrate its correctness. In fact, we have tried a lot of examples by Magma, the conjecture is also correct. But we fail to prove it. Thus we would like to put it here as a conjecture.

Example 3.6 Let R

k

= F

3

[ u ] / u

^k

, m = 11, α = 2 be an even integer, implies α | (m − 1) = 2 | 10,ord

₁₁

(3) =

^m−1_α

= 5, then by Conjecture 3.5,

x

¹¹

− 1 = (x − 1)(x

⁵

+ 2x

³

+ x

²

+ 2x + 2)(x

⁵

+ x

⁴

+ 2x

³

+ x

²

+ 2), x

¹¹

+ 1 = (x + 1)(x

⁵

+ 2x

³

+ 2x

²

+ 2x + 1)(x

⁵

+ 2x

⁴

+ 2x

³

+ 2x

²

+ 1).

Example 3.7 Let R

k

= F

2

[ u ] / u

^k

, m = 5, u

^k

= 0, t ≥

^k₂

, α = 1 be an odd integer, implies α | (m − 1) = 1 | 4,ord

₅

(2) =

^m−1_α

= 4, then by Conjecture 3.5,

x

⁵

− 1 = (x − 1)(x

⁴

+ x

³

+ x

²

+ x + 1),

x

⁵

− (1 + u

^t

) = (x + 1 + u

^t

)(x

⁴

+ (1 + u

^t

)x

³

+ x

²

+ (1 + u

^t

)x + 1).

Example 3.8 Let R

_k

= F

4

[ u ] / u

^k

, m = 7, u

^k

= 0, t ≥

^k₂

, α = 2 be an even integer, implies α | (m − 1) = 2 | 6,ord

7

(4) =

^m−1_α

= 3, then by Conjecture 3.5,

x

⁷

− 1 = (x − 1)(x

³

+ x + 1)(x

³

+ x

²

+ 1),

x

⁷

− (1 + u

^t

) = (x + 1 + u

^t

)(x

³

+ x + 1 + u

^t

)(x

³

+ (1 + u

^t

)x

²

+ 1 + u

^t

).

The proof of Theorem 3.9 is similar to that of Theorem 3.2, and is omitted.

Theorem 3.9 Assume that the factorization of x

^m

− λ into basic irreducible polynomials over R

_k

=

^F_u^q[u]k

is of the form of

1) case (1) in Conjecture 3.5, the total number of self-dual (resp. LCD) double λ-circulant codes over R

_k

is

Bq

^{(m−1)(k−1)2}

(q

^m−12α

+ 1)

^α

(resp.(q − 2)q

^m(k−1)

(q

^m−1α

− (q

^m−12α

+ 1))

^α

).