Some trace-restricted cyclic and constacyclic MDS symbol-pair codes

Introduction

Coding theory is concerned with mathematical techniques for the reliable transmission and storage of information over noisy communication channels. During the past decades, it has outgrown into a broad area of research with applications in distributed storage, digital communication, wireless networks, information security, and optical communication. The classical theory of error-correcting codes [16] is derived mostly from the Hamming metric [1] , that assumes symbols are read individually in the decoding process. In various high-density storage systems, the reading mechanism is lower than the writing mechanism and information is accessed through overlapping pairs of adjacent symbols. This observation results to the introduction of the symbol-pair metric [2] , that gives a more relevant model for such channels and has generated vigrous research on the construction and exploration of symbol-pair codes. Among several methods, cyclic and constacyclic codes have received vast attention because of their rich algebraic structure and streamlined implementation. Several families of optimal symbol-pair codes have been derived by implementing BCH-type techniques, defining sets, constacyclic structures over finite fields and repeated-root cyclic codes. These findings have demonstrated close connections between the defining roots and the symbol-pair distance of the resulting codes. In the classical setting, extension fields and Frobenius conjugacy [19,20] are employed to determine generator polynomials over the base field, while the arrangement of consecutive defining roots results lower bounds on the minimum Hamming or symbol-pair distance. Hence, various constructions of MDS symbol-pair codes depend on the attentive selection of defining roots in finite field extensions. Although defining-root constructions over finite fields have been extensively studied, very few studies have explored constructions over finite rings with nilpotent elements. Finite rings have various algebraic features unlike fields, including richer ideal structures and nontrivial nilpotent components, making them an interesting setting for the construction of linear codes offering additional algebraic flexibility which suggests the possibility of introducing definingroot techniques beyond the classical finite-field framework. Also, the interaction between trace conditions, defining roots, and the symbol-pair metric over these rings has not been fully investigated. So trace-restricted roots are introduced to construct these codes whose generator polynomials are defined over the base field. The corresponding symbol-pair distance is then analyzed through the arrangement of consecutive defining roots, yielding conditions under which the resulting codes attain the Singletontype bound in the symbol-pair metric. The proposed approach combines trace techniques with finite-ring methods to obtain several families of cyclic and constacyclic MDS symbol-pair codes.

Preliminaries

Symbol-pair read vector [2]: Let 𝑐 = (𝑐 0 , 𝑐 1 , … , 𝑐 𝑛-1 ) ∈ 𝔽 𝑞 𝑛 .

The symbol-pair read vector of 𝑐 is defined by

where the indices are taken modulo 𝑛.

Symbol-pair distance [2]: Let 𝑐, 𝑐 ′ ∈ 𝔽 𝑞 𝑛 . The symbol-pair distance between 𝑐 and 𝑐 ′ is defined by

where 𝑑 𝐻 denotes the Hamming distance. The minimum symbol-pair distance of a linear code 𝒞 ⊆ 𝔽 𝑞 𝑛 is

Cyclic and constacyclic codes [16] :

When 𝛾 = 1, the code is called a cyclic code.

Frobenius Automorphism [20]: Let 𝔽 𝑞 𝑚 be an extension field of 𝔽 𝑞 . The map

is called the Frobenius automorphism. Its iterates are given by 𝜎 𝑖 (𝑥) = 𝑥 𝑞 𝑖 , 𝑖 ≥ 0.

MDS symbol-pair code [11]: Let 𝒞 be an [𝑛, 𝑘] linear code over 𝔽 𝑞 . By the Singleton-type bound for the symbol-pair metric,

If equality holds, then 𝒞 is called a maximum distance separable (MDS) symbol-pair code.

Trace-Restricted Defining Roots over the Ring 𝑹

Establishing the finite commutative ring together with the trace mapping Type equation here.which makes the base of the defining-root construction for that the algebraic properties are introduced and used to construct cyclic and constacyclic symbolpair codes. Let

Every element of 𝑅 has a unique representation

where 𝑥 𝑖 in 𝔽 𝑞 for 0 ≤ 𝑖 ≤ 5. Remark 1. The ring 𝑅 is a finite commutative ring containing nilpotent elements of index three. These nilpotent components enrich the algebraic structure of 𝑅 and provide additional flexibility for constructing defining roots beyond the classical finitefield setting.

Lemma 1. Let 𝒯 𝑅: 𝑅 → 𝑅 be the trace mapping. Then the following properties hold for all 𝑥, 𝑦 ∈ 𝑅 and 𝑎 ∈ 𝔽 𝑞: 1. 𝒯 𝑅 (𝑥 + 𝑦) = 𝒯 𝑅 (𝑥) + 𝒯 𝑅 (𝑦). Since 𝑎 𝑞 = 𝑎 for every 𝑎 ∈ 𝔽 𝑞 ,

establishing 𝔽 𝑞 -linearity.

Finally, the kernel of every linear mapping is an 𝔽 𝑞 -subspace of its domain.

Construction over the Nilpotent Chain Ring

Let 𝑅 = 𝔽 𝑞 + 𝑢𝔽 𝑞 + 𝑣𝔽 𝑞 + 𝑢 2 𝔽 𝑞 + 𝑢𝑣𝔽 𝑞 + 𝑣 2 𝔽 𝑞 be the finite commutative ring, where 𝑢 3 = 𝑣 3 = 0 and 𝑢𝑣 = 𝑣𝑢.

Now,

𝛼 𝑞 -𝛼 = (𝑎 𝑞 -𝑎) + 𝑢(𝑏 𝑞 -𝑏) + 𝑣(𝑐 𝑞 -𝑐) + 𝑢 2 (𝑑 𝑞 -𝑑) + 𝑢𝑣(𝑒 𝑞 -𝑒) + 𝑣 2 (𝑓 𝑞 -𝑓).

Since 𝑢 3 = 𝑣 3 = 0, 𝑢 2 𝑣 = 𝑢𝑣 2 = 0, it follows that 𝑢(𝛼 𝑞 -𝛼) = 𝑢(𝑎 𝑞 -𝑎) + 𝑢 2 (𝑏 𝑞 -𝑏) + 𝑢𝑣(𝑐 𝑞 -𝑐), and 𝑣(𝛼 𝑞 -𝛼) = 𝑣(𝑎 𝑞 -𝑎) + 𝑢𝑣(𝑏 𝑞 -𝑏) + 𝑣 2 (𝑐 𝑞 -𝑐).

Hence, 0 = (𝑎 + 𝑎 𝑞 ) +𝑢(𝑏 + 𝑏 𝑞 + 𝑎 𝑞 -𝑎) +𝑣(𝑐 + 𝑐 𝑞 + 𝑎 𝑞 -𝑎) +𝑢 2 (𝑑 + 𝑑 𝑞 + 𝑏 𝑞 -𝑏) +𝑢𝑣(𝑒 + 𝑒 𝑞 + 𝑏 𝑞 -𝑏 + 𝑐 𝑞 -𝑐) +𝑣 2 (𝑓 + 𝑓 𝑞 + 𝑐 𝑞 -𝑐).

Since {1, 𝑢, 𝑣, 𝑢 2 , 𝑢𝑣, 𝑣 2 } forms an 𝔽 𝑞 -basis of 𝑅, each coefficient must vanish individually, yielding the required system of equations.

Lemma 3. Let 𝛼 𝑖 ∈ 𝑅 be a root satisfying 𝒯 𝑅 (𝑖) = 0. Then its Frobenius conjugate 𝛼 𝑖 𝑞 also satisfies 𝒯 𝑅 (𝑖 𝑞 ) = 0 and the pair (𝛼 𝑖 , 𝛼 𝑖 𝑞 ) generates a quadratic factor in 𝔽 𝑞 [𝑥].

Proof. From above lemmas the trace condition implies 𝛼 𝑖 𝑞 = -𝛼 𝑖 modulo the nilpotent component. Hence 𝒯 𝑅 (𝑖 𝑞 ) = 𝑖 𝑞 + (𝑖 𝑞 ) 𝑞 + 𝑢((𝑖 𝑞 ) 𝑞 -𝑖 𝑞 ) = 𝑖 𝑞 -𝑖 𝑞 + 𝑢(0) = 0.

The minimal polynomial over 𝔽 𝑞 therefore contains both 𝛼 𝑖 and 𝛼 𝑖 𝑞 and has degree at most two. Since neither element lies in 𝔽 𝑞 , the degree equals two.

Families of Code Length Arising from Trace-Restricted Nilpotent Roots

Let 𝑅 be the nilpotent chain ring introduced earlier, and let 𝒯 𝑅 (⋅) denote the trace map. Let the code length is denoted by 𝑛 and vectors belong to 𝑅 𝑛 . All symbols follow the same notation used in the previous sections. The purpose of this section is to describe several families of length parameters obtained from consecutive trace-zero roots 𝛼 𝑖 inside the multiplicative orbit generated by a primitive element of 𝑅. The arguments rely on the Frobenius action together with the nilpotent structure of the ring.

Length Family 𝒏 = 𝟑𝒒 + 𝝁

Assume that 𝑛 = 3𝑞 + 𝜇 with 𝜇 ≥ 5. Let 𝜉 ∈ 𝑅 generate a cyclic orbit {𝜉 0 , 𝜉 1 , … , 𝜉 𝑛-1 }. Let So 𝜆 -2 consecutive roots are there in defining set. Hence the pair metric yields 𝑑 p (𝒞 𝜆 ) ≥ 𝜆.

for any code in 𝑅 𝑛 . Both inequalities coincide, therefore equality holds. ◻

Length Family 𝒏 = 𝒒 𝟐 -𝒕

Let 𝜉 be a primitive element of order 𝑛 and 𝑛 = 𝑞 2 -𝑡 with 1 ≤ 𝑡 ≤ 𝑞. Let 𝛼 𝑖 = 𝜉 𝑖 whenever 𝒯 𝑅 (𝜉 𝑖 ) = 0.

Lemma 5 [29] . Let 𝜉 be a primitive 𝑁th root of unity, where 𝑁 = 𝑞 2 -𝑡, 𝑁 | (𝑞 2 -1). If 𝜆 ≤ 𝑁 2 + 2, then there exists an integer 𝑖 such that the consecutive roots 𝜉 𝑖 , 𝜉 𝑖+1 , … , 𝜉 𝑖+𝜆-3 all belong to the trace-zero subset of 𝑅.

Proof. By the result established in [29] , the trace-zero elements arising from the orbit of a primitive 𝑁 𝑡ℎ root of unity form a subset that contains a consecutive block of length at least 𝜆 -2 whenever , where 𝒩(ϑ) = ϑϑ q is the Frobenius norm. The cyclic code X λ has dimension K = N -(λ -2). Its minimum symbol-pair distance satisfies d pair (X λ ) ≥ λ.

Applying the Frobenius automorphism

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