Polyadic cyclic codes or simply called polyadic codes form an important class of cyclic codes. They have rich algebraic structures for efficient error detection and correction, which explains their preferred role in engineering. Polyadic codes generalize quadratic residue codes, duadic codes, triadic codes and m-adic residue codes.
Codes over finite rings have been known for several decades, but interest in these codes increased substantially after a break-through work by Hammons et al. in 1994, which shows that some well known binary non-linear codes like Kerdock codes and Preparata codes can be constructed from linear codes over . Since then, a lot of research has been done on cyclic codes, in particular on quadratic residue codes over different types of finite rings such as integer residue rings , Galois rings , chain rings and non-chain rings. Kaya et al.  and Zhang et al.  studied quadratic residue codes over a non-chain ring , where and p is an odd prime. Bayram and Siap  considered cyclic and constacyclic codes over , where p is a prime. Kaya et al.  studied quadratic residue codes over , whereas Liu et al.  studied them over non-local ring , where and p is an odd prime. The authors  along with Kathuria extended their results over the ring , where and . In , the authors studied quadratic residue codes and their extensions over the ring , where , m any integer greater than 1 and p is a prime satisfying . In , the authors studied duadic codes over the ring , where q is a prime power satisfying . In , the authors considered a more general non-chain ring , where q is a prime power and is a polynomial of degree , which splits into distinct linear factors over and studied duadic and triadic codes over it generalizing all the previous results. In another paper , the authors have studied duadic negacyclic codes over the ring . In , Kuruz et al. studied m-adic residue codes over .
Recently people have started studying codes over finite commutative non-chain rings having 2 or more variables. Ashraf and Mohammad  studied cyclic codes over . They  also studied skew-cyclic codes over , where . Srinivasulu and Bhaintwal  studied linear codes over , where , a non-chain extension of . Yao, Shi and Solé  studied skew cyclic codes over , where and q is a prime power. Islam and Prakash  studied skew cyclic and skew constacyclic codes over , where and . Note that all the polynomials considered namely or with etc. split into distinct linear factors over .
In this paper, we study cyclic and polyadic cyclic codes over a more general ring. Let and be two non-constant polynomials of degree k and respectively (k and not both 1), which split into distinct linear factors over . Let be a finite commutative non-chain ring. Here we discuss cyclic codes and their duals over the ring , define polyadic codes over in terms of idempotent generators and study some of their properties. We also give examples of some codes that have optimal parameters with respect to Griesmer type bound for rings. A Gray map is defined from which preserves linearity and in some special case preserves duality. The Gray images of polyadic codes over the ring and their extensions lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over .
The paper is organized as follows: In Section 2, we give some preliminaries including Griesmer type bound for codes over rings, recall polyadic codes of length n over and give some of their properties. In Section 3, we study the ring , cyclic codes over ring and define the Gray map : . In Section 4, we study polyadic codes over , their extensions, their Gray images and discuss Griesmer type bound for these codes. We give some examples to illustrate our theory.
A cyclic code of length n over can be regarded as an ideal of the ring . It has a unique generating polynomial and a unique idempotent generator . The set , where is a primitive nth root of unity in some extension field of , is called the defining set of .
A polynomial is called even-like if otherwise it is called odd-like. A code is called even-like if all its codewords are even-like otherwise it is called odd-like.
For , defined as is called a multiplier, where . It is extended on by defining .
For a linear code over , the dual code is defined as , where denotes the usual Euclidean inner product. is self-dual if and self-orthogonal if . A code is called isodual if it is equivalent to its dual . A linear code whose dual satisfies is called a complementary dual (LCD) code.
Let . The even weight cyclic code over has generating idempotent , its dual is the repetition code with generating idempotent .
The following is a well known result, see :
Lemma 1: 1) Let be a cyclic code of length n over a finite field with defining set T. Then the defining set of is and that of is .
2) Let and be cyclic codes of length n over a finite field with defining sets and respectively. Then and are cyclic codes with defining sets and respectively.
3) Let and be cyclic codes of length n over generated by the idempotents in , then and are generated by the idempotents and respectively.
4) Let be a cyclic code of length n over generated by the idempotent E, then is generated by and is generated by the idempotent .
A linear code over a finite commutative ring R is an R-submodule of . Dual of a linear code over a finite commutative ring is defined in the same way and results in Lemma 1 (3) and (4) also hold true over any finite ring.
2.1. Griesmer Type Bound for Codes over Rings
Let R be a finite commutative quasi-Frobenious ring. For a linear code C over R, the value is defined as the rank of minimal free R-submodules of which contain C. Let , where are central orthogonal idempotents with, . Then is also a QF ring for each . Let denote the Jacobson radical of R. If C is a linear code of length n over R, then is a linear code of length n over .
The following Griesmer type bound is due to Shiromoto and Storme ( , Theorem 2.6).
Theorem 1: Let be a finite quasi-Frobenious ring such that is a local ring for all and let be the prime power such that for each . If C is a linear code of length n over R, then
where , and .
The code C over R is said to have parameters .
2.2. Polyadic Cyclic Codes over
Let and suppose
1) and are union of q-cyclotomic cosets mod n,
2) and are pairwise disjoint,
3) there exists a multiplier , such that , for , the subscripts are taken modulo m and .
It is clear that always. Let .
Then codes, for , having or as their defining sets are called odd-like polyadic codes and the codes having or as their defining sets are the associated even-like polyadic codes. Let denote the odd-like codes having as their defining sets; denote the odd-like codes having as their defining sets; denote the even-like codes having as their defining sets; and denote the even-like codes having as their defining sets.
In the special case, when and , polyadic codes are duadic codes . When , polyadic codes are triadic codes as defined by Pless and Rushanan . When , an odd prime, , , , , , then polyadic codes are m-adic residue codes as defined by Job . A polyadic code of prime length p exists if and only if q ia an m-adic residue mod p, see Brualdi and Pless . When n is a prime power, the conditions for the existence of polyadic codes over were obtained by Sharma et al.  and for general n see Bakshi et al. .
Clearly are equivalent codes; are equivalent; are equivalent; and are equivalent codes.
For , let and be the even-like idempotent generators of even-like polyadic codes and respectively, and be odd-like idempotent generators of odd-like polyadic codes and respectively.
As the defining set of is , the defining set of is . Therefore and hence . Similarly, for and for . Similar results hold for and .
Let the set be denoted by A. Similar to the properties of triadic codes obtained in , we have the following results for polyadic codes over .
Proposition 1: For any subset , where , we have
5) , for ,
10) , for .
Proof: By Lemma 1 (2), the defining set of each of , is , hence they are equal. The defining set of is , which is the defining set of even weight code having generating idempotent . Again by Lemma 1 (2), the defining set of each of , is , hence they are all equal. The defining set of is , which is the defining set of the repetition code having generating idempotent . The defining set of is whole of , hence it is in the ring ; whereas defining set of is , so it is . The other results follow by Lemma 1 (3).
Proposition 2: For any subset , where , we have
5) , for ,
10) , ,
11) , ,
12) , ,
13) , ,
14) , ,
Proof: Statements (1) to (10) are similar to those of (1) to (10) of Proposition 1. For (11), we note that the defining set of is . Therefore the defining set of is and defining set of is same as that of . Similarly we have (12). The defining set of is and that of is . The defining set of is and that of is . Now (15) and (16) follow by Lemma 1(3).
Proposition 3: Suppose is empty, then for any subset , where , we have the following additional results:
Proof is straightforward.
Proposition 4: Let , , for , be two pairs of even-like polyadic codes over with , the associated pairs of odd-like polyadic codes. Then and .
Further if , then , and so , , and are LCD codes.
Proof: As the defining set of is , the defining set of , by Lemma 1 (1) is
This proves that . Similar is the proof of others. When , we get, from Propositions 1 (5) and 2 (5), that and ; proving that and are LCD codes. One can check that and are also LCD codes.
3. Cyclic Codes over the Ring and the Gray Map
3.1. The Ring
Let q be a prime power, . Throughout the paper, denotes the commutative ring , where and are two non-constant polynomials of degree k and respectively, which split into distinct linear factors over . We assume that k and are not both 1, otherwise . If or , then the ring is isomorphic to or . Cyclic, duadic and triadic codes over have been discussed by the authors in .
Let , with , and , with , . is a non chain ring of size and characteristic p.
For and , let , and , , be elements of the ring given by
If , we define and if , we take .
For , define as follows
Lemma 2: We have , for , , and in , i.e., ’s are primitive orthogonal idempotents of the ring .
Proof: Since for and for , in . To prove , it is enough to prove that in . For that we need to prove for all r and for all s. If , then . If , then , hence , for and , for . One can easily check that and , so in and hence in .
Now to prove in , it is sufficient to prove that . This can be easily checked as and for all r and s, .
The decomposition theorem of ring theory tells us that .
For a linear code of length n over the ring , let for each pair , , let
Then are linear codes of length n over , and .
The following is a simple generalization of Theorem 1 of .
Theorem 2: Let be a linear code of length n over . Then
1) is cyclic over if and only if are cyclic over .
2) If , , then,
where and .
3) Further .
4) Suppose that . Let , then .
6) , , where is the reciprocal polynomial of .
3.2. The Gray Map
Every element of the ring can be uniquely expressed as
where for .
Define a Gray map by
where V is any nonsingular matrix over of order . This map can be extended from to component wise.
Let the Gray weight of an element be , the Hamming weight of . The Gray weight of a codeword is defined as . For any two elements , the Gray distance is given by .
Theorem 3: The Gray map is an —linear, one to one and onto map. It is also distance preserving map from ( , Gray distance ) to ( , Hamming distance ). Further if the matrix V satisfies , , where denotes the transpose of the matrix V, then for any linear code over .
Proof. The first two assertions hold as V is an invertible matrix over .
Let now , where is a row vector, satisfying . So that
Let be a linear code over . Let , , where and . So that . It is enough to prove that . Using the properties of ’s from Lemma 2, we get
Using (5) and (6), we find that
which proves the result.
4. Polyadic Codes over the Ring
We now define polyadic codes of length n over the ring in terms of their idempotent generators with the assumption that the conditions on n and q for existence of polyadic codes over the field are satisfied. Let be idempotents as defined in (3) and (4). Let the set of ordered suffixes be divided into m disjoint subsets
with the assumption that each of the sets is non-empty, if . In that case let .
If , we assume that in the partition (7), sets are non-empty, each containing exactly one element and the remaining sets are empty. Therefore , if is non empty and , if is empty.
with the convention that empty sum is regarded as zero.
Using Lemma 2, we find that
and that are mutually orthogonal idempotents in the ring , i.e.,
For , let be the idempotent generators of polyadic codes over as defined in Section 2.2.
For each tuple , let
be odd-like idempotents in the ring . Similarly let
be even-like idempotents in the ring .
For each tuple , and for each , let denote the odd-like polyadic codes and denote the even-like polyadic codes over generated by the corresponding idempotents, i.e.
Clearly for any tuple , are equivalent; are equivalent; are equivalent and are equivalent.
Next we compute the number of inequivalent odd-like and even-like polyadic codes over the ring .
Theorem 4: If , then there are
inequivalent odd-like polyadic codes and the same number of inequivalent even-like polyadic codes over the ring .
If , then there are
inequivalent odd-like polyadic codes and the same number of inequivalent even-like polyadic codes over the ring .
Proof: Let first , out of idempotents , can be chosen in ways. Out of remaining idempotents can be chosen in ways, continuing like this can be chosen in ways and will be fixed. As each must have at least one , the number of choices of idempotents is .
Since , and ’s contribute equal number of inequivalent odd-like idempotents, we get the desired number.
Let now . Firstly the non-empty sets in the partition (7) can be chosen in ways. Out of idempotents , first non-zero can be chosen in ways, next non-zero can be chosen in ways, , so the number of choices of is . Since , and ’s contribute equal number of inequivalent odd-like idempotents, we get the required number.
We drop the superscript , when there is no confusion with the idempotents or the corresponding polyadic codes.
Theorem 5: For any subset , where , the following assertions hold for polyadic codes over .
1) , the repitition code over
4) , the even weight code over ,
6) , .
Proof: From the definitions and relations (8)-(11), we find that the sums of products of terms from taken one at a time, taken two at a time and so on is equal to the sums of products of terms from taken one at a time, taken two at a time and so on, i.e.
Therefore by Lemma 1 and relations (13), we find that
By Proposition 1 (8) and (9), we get (1) and (2).
To prove (3), from Proposition 1 (6) we see that
Similarly for any tuple . Hence , so we get (3) by Lemma 1.
Again as and for any tuple , , we see that
Now (4) follows from Proposition 1 (7).
Since for all by Proposition 2 (15), we get and so . As , we find that and so . Therefore . This proves (5).
We prove (6) for . Others are similar. Note that and , by Proposition 1 (10). Therefore and .
Similarly we have.
Theorem 6: For any subset , where , the following assertions hold for polyadic codes over .
5) , ,
6) , ,
7) , ,
8) , and,
9) , .
Proof: The proof of statements (1) to (6) is similar to that of (1) to (6) of Theorem 5. To prove (7) we note that
by Proposition 2 (15).
Hence . Similarly others. Statements (8) and (9) follow from Proposition 2 (16).
Theorem 7: Let , , for , be two pairs of even-like polyadic codes over the ring with , the associated pairs of odd-like polyadic codes. Then
Further if for , then
, and , , , are LCD codes over .
Proof: By Proposition 1 (10), . So . Therefore
Hence . Similarly, we get the others.
Further if for , then by Theorem 5 (6) and Theorem 6 (6),
proving thereby that and are LCD codes over . Similarly one can check that and are also LCD codes over .
Theorem 8: If is empty, then we have the following additional results:
1) , .
2) , .
Proof: Here since , by Proposition 3, we have for ant tuple . Therefore for any s, , and . Hence by proposition 5 (4),
As , we get that . Since from Theorems 5 and 6, we have and ,
Again from Theorem 6 (8), we see that , which gives . Finally gives
4.1. Extensions of Polyadic Codes over the Ring
When is empty, we consider extended polyadic codes over the ring which give us some additional results.
Consider the equation
This equation has a solution in if and only if n and -1 are both squares or both non squares in (see , Chapter 6).
For a linear code C of length n over , , the extension of C is defined as
Theorem 9: Let be empty. Suppose there exists a in satisfying Equation (13). If the splitting of in (1) is given by the multiplier , then the extended odd-like polyadic codes satisfy .
Proof: Here, by Theorem 7, . As and , by Theorem 6 (7), let and be the extended polyadic code over generated by
where is a generator matrix for the even-like polyadic code and is a generator matrix for the even-like polyadic code . The row above the matrix shows the column labeling by . Since the all one vector belongs to and its dual is equal to , the last row of is orthogonal to all rows of . The last row is orthogonal to itself also as in . Therefore all rows of are orthogonal to all the rows of . Now the result follows from the fact that , as can be verified from Theorem 8.
Similarly, we have
Theorem 10: Let be empty. Suppose there exists a in satisfying Equation (14). If so that the splitting of in (2) is not given by the multiplier , then the extended odd-like polyadic codes satisfy .
Corollary 1: If is empty, then the following assertions hold for duadic codes over .
1) If , then ; are self orthogonal and are self dual.
2) If then are isodual.
Proof: Here, by definition and , therefore , , , , and .
If , , i.e., when the splitting is given by , we have by Theorem 7, , subscript modulo m. Therefore . Using statement (7) of Theorem 6 , we have . Therefore is self-orthogonal. By Theorem 9, , therefore .
If , By Theorem 10, , therefore and .
4.2. Griesmer Type Bound for Polyadic Codes over
Kuruz et al.  gave some examples of m-adic residue codes over whose parameters attain Griesmer type bound. In the next theorem, we prove that the Griesmer type bound for polyadic codes over the ring is same as the Griesmer bound for the corresponding polyadic codes over the field .
Theorem 11: The parameters of polyadic codes over are same as parameters of the corresponding polyadic codes over . Hence Griesmer type bound for polyadic codes over the ring is same as the Griesmer bound for the corresponding polyadic codes over the field .
Proof: Let be a polyadic code of length n over . Then is equal to or or or for . By definition, , where are all odd-like polyadic codes over and are equivalent. Therefore by Theorem 1,
Further . Here Jacobson radical, , so for every i andj. Hence the Griesmer type bound for odd-like polyadic codes over the ring becomes
which is same as Griesmer bound for polyadic code over .
Similar result holds for , and .
Example 1: Let , , , and . Take . Here has parameters . It attains the Griesmer type bound over the ring . Therefore are optimal. The code has parameters . It nearly attains Griesmer type bound over the ring .
Example 2: Let , , , and . Here has parameters , so it attains the Griesmer type bound over the ring . Therefore and are optimal.
Remark: Using the above theory, one can construct some other cyclic codes over the ring (which are not polyadic according to our definition) generated
by idempotents of the type
where are subsets of , and which may attain the Griesmer type bound.
For example, take , , , , and . Let C be a cyclic code over ring generated by the idempotent E, then C has parameters and it attains the Griesmer type bound.
As an another example, take , , , and and . The cyclic code C generated by the idempotent over ring has parameters and it nearly attains the Griesmer type bound.
4.3. Gray Images of Polyadic Codes over
Theorem 12: Let the matrix V taken in the definition of the Gray map satisfy , . For all possible choices of , the Gray images of even-like polyadic codes and Gray images of extensions of odd-like polyadic codes , for , have the following properties
1) If , i.e., if , then , , and are linear complementary dual(LCD) codes of length over .
2) If is empty and , then .
3) If is empty and , i.e., if the splitting in (1) is given by then .
The theorem follows from Theorems 3, 7, 9 and 10.
Corollary 2: If is empty, , then the following assertions hold for duadic codes over .
1) If , then are self orthogonal of length and are self-dual codes of length over .
2) If , then are isodual codes of length over .
The following examples illustrate our theory. The minimum distances of all these codes have been computed by the Magma Computational Algebra System.
Example 3: Let , , , , , and
be a matrix over satisfying . Here , and . Also and . On taking and , we have and . The Gray images of polyadic codes and are self-orthogonal [18, 6, 6] and self-dual [24, 12, 4] codes over respectively.
Example 4: Let , , , , and
be a matrix over satisfying , where a is a primitive element of . The Gray images of polyadic codes , , and with and are respectively [68, 16, 28], [68, 52, 6], [68, 48, 8] and [68, 20, 17] LCD codes over .
Some other examples of self-dual, self-orthogonal and LCD codes arising as Gray images of Polyadic codes over are given in Table 1. The matrices A and B in Table 1 are as defined in Examples 3 and 4 respectively and the matrix C is taken as
*In this case, .
†H4 is Hadamard matrix of order 4.
‡I is the Identity matrix.
Table 1. Gray images of some polyadic codes.
In this paper, polyadic codes and their extensions over a finite commutative non-chain ring are studied where and are two polynomials of degree k and respectively (k and are not both 1) which split into distinct linear factors over . A Gray map is defined from which preserves duality. As a consequence, self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over are constructed. Some examples are also given to illustrate our theory. It is shown that the Griesmer type bound for polyadic codes over the ring is same as the Griesmer bound for the corresponding polyadic codes over the field . Examples of some codes which are optimal with respect to Griesmer type bound are given. The results of this paper can easily be extended over the ring where polynomials , , split into distinct linear factors over .
The second author gratefully acknowledges support from Council of Scientific & Industrial Research (CSIR), India, Sanction No. 21(1042)/17/EMR-II.
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