Parameters of several families of binary duadic codes and their related codes
Hai Liu, Chengju Li, Haifeng Qian

TL;DR
This paper explores various families of binary duadic codes, providing new lower bounds on their minimum distances and analyzing their dual and extended code parameters, advancing understanding of their error-correcting capabilities.
Contribution
It introduces several new families of binary duadic codes with bounds on minimum distances and investigates their dual and extended code parameters.
Findings
Lower bounds on minimum distances are close to the square-root bound.
Parameters of dual and extended codes are characterized.
Partially solves an open problem on minimum distance bounds.
Abstract
Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes, which partially solves one case of the open problem proposed in \cite{LLD}. It is shown that the lower bounds on their minimum distances are close to the square root bound. Moreover, the parameters of the dual and extended codes of these binary duadic codes are investigated.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Cancer Mechanisms and Therapy
Parameters of several families of binary duadic codes and their related codes
††thanks: The work of Hai Liu and Chengju Li was supported by the National Natural Science Foundation of China (12071138), Shanghai Natural Science Foundation (22ZR1419600), the open research fund of National Mobile Communications Research Laboratory of Southeast University (2022D05). The work of Haifeng Qian was supported by the Innovation Program of Shanghai Municipal Education Commission (2021-01-07-00-08-E00101), and “Digital Silk Road” Shanghai International Joint Lab of Trustworthy Intelligent Software (22510750100).
Hai Liu, Chengju Li, Haifeng Qian H. Liu and C. Li are with MoE Engineering Research Center of Software/Hardware Co-design Technology and Application, East China Normal University, Shanghai, 200062, China; and are also with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (email: [email protected], [email protected]).H. Qian is with the School of Software Engineering East China Normal University, Shanghai 200062, China (email: [email protected]).
Abstract
Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes, which partially solves one case of the open problem proposed in [10]. It is shown that the lower bounds on their minimum distances are close to the square root bound. Moreover, the parameters of the dual and extended codes of these binary duadic codes are investigated.
1 Introduction
In this paper, let denote the finite field of order , where is a power of a prime . An linear code over is a -dimensional subspace of with minimum (Hamming) distance . The dual code of , denoted by , is defined by
[TABLE]
where is the standard inner product of two vectors and in . In addition, define the extended code to be the code
[TABLE]
It is easy to see that is an linear code.
The linear code over is said to be cyclic if implies . By identifying each vector with
[TABLE]
a code of length over corresponds to a subset of . Then is a cyclic code if and only if the corresponding subset is an ideal of . Note that every ideal of is principal. Then there is a monic polynomial of the smallest degree such that and . Then is called the generator polynomial and is referred to as the check polynomial of . Throughout this paper, assume that . Denote , i.e., is the smallest positive integer such that . Let be a primitive element of and put . Then is a primitive -th root of unity. The set is referred to as the defining set of with respect to . If contains consecutive integers, then we have the well-known BCH bound on the minimum distance of cyclic codes, i.e., .
Let and be two subsets of such that
- •
and , and
- •
both and are the union of some -cyclotomic cosets modulo .
If there is a unit such that and , then is called a splitting of .
Let be a splitting of . Define
[TABLE]
for . The pair of cyclic codes and of length over with generator polynomials and are called odd-like duadic codes, and the pair of cyclic codes and of length over with generator polynomials and are called even-like duadic codes.
By definition, the binary cyclic codes and have parameters and the binary cyclic codes and have parameters . Binary cyclic codes with parameters were investigated in [2, 7, 12, 13, 14]. It is observed that these binary cyclic codes have large dimensions. Generally, it is very hard to determine the minimum distance of a cyclic code with parameters . For now, the best one can do is to develop a good lower bound on the minimum distance of the code. For odd-like duadic codes, we have the following result [7, Theorem 6.5.2].
Theorem 1.1** (Square root bound).**
Let and be a pair of odd-like duadic codes of length over . Let be their (common) minimum odd weight. Then the following hold:
. 2. 2.
If the splitting defining the duadic codes is given by , then . 3. 3.
Suppose , where , and assume that the splitting defining the duadic codes is given by . Then is the minimum weight of both and .
As a generalisation of the quadratic residue codes, duadic codes were introduced and investigated in [8, 9, 11], where a number of properties are proved. In addition, Pless, Masley and Leon presented all binary duadic codes of length until 241 [11]. The total number of binary duadic codes of prime power lengths and their constructions were presented in [4] and [5]. For more information on duadic codes, the reader is referred to [7, Chapter 6].
Let be a positive integer and let . For any , the -cyclotomic coset of modulo is defined by
[TABLE]
where is the smallest positive integer such that . For an integer with , let
[TABLE]
be the -adic expansion of , where with . For any with , define .
Now we recall a construction of binary duadic codes documented in [10]. Let be a positive integer and let for an integer . Let be any proper subset of . Define
[TABLE]
By definition, is the union of some -cyclotomic cosets modulo . Let be a primitive element of . Let denote the binary cyclic code of length with generator polynomial
[TABLE]
Let be even and , the code could be a duadic code for certain odd .
When and , a family of duadic codes were studied in [12]. 2. 2.
When and , two families of duadic codes were presented in [10], where the following open problem was also proposed.
Open problem. [10] Let be an even integer. Find a subset of with such that is a binary duadic code of length for infinitely many odd . Determine the parameters of these duadic codes.
In this paper, we present binary duadic codes for all possible when and , and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes. As will be seen, these lower bounds are close to the square root bound. This partially solves one case of the open problem. Moreover, the parameters of the dual and extended codes of these binary duadic codes are also investigated.
2 Constructions of all binary duadic codes for
In this section, we follow the notation specified in Section 1. Let be any proper subset of with and . It is clear that is the defining set of with respect to the -th primitive root of unity , where . Note that . It then follows that for a splitting of . Moreover, we have . Then all binary duadic codes for can be constructed as follows.
When , let S\in\Big{\{}\{0,2,3\},\{0,2,4\},\{0,4,5\},\{0,3,5\}\Big{\}}. Then and . Note that for each with . This leads to
[TABLE]
Thus and form a pair of odd-like duadic codes. 2. 2.
When , let S\in\Big{\{}\{0,1,4\},\{0,1,5\},\{0,2,4\},\{0,2,5\}\Big{\}}. Then one can verify that and form a pair of odd-like duadic codes. 3. 3.
When , let S\in\Big{\{}\{0,1,2\},\{0,1,3\},\{0,2,4\},\{0,3,4\}\Big{\}}. Then one can verify that and form a pair of odd-like duadic codes.
It is remarked that the parameters of the codes and for had been studied in [12]. Below we mainly investigate the parameters of the binary duadic codes and except and their dual and extended codes. Lower bounds on their minimum distances are presented, and they are close to the square root bounds.
3 Some auxiliary results
In this section, we will present some necessary auxiliary results on the defining sets of binary duadic cyclic codes, which play an important role in developing lower bounds on minimum distances of the binary codes. The following well-known lemma will be employed later.
Lemma 3.1**.**
Let and be two positive integers. Then
[TABLE]
where is a positive integer.
Below we divide into three cases according to the construction of the binary duadic codes.
A. The case:
When , we have two subcases: and .
Lemma 3.2**.**
Let . Then we have the following.
If , then and
[TABLE] 2. 2.
If , then and
[TABLE]
Proof.
If , it follows from Lemma 3.1 that . When , we have
[TABLE]
Consequently, . When , and . When , . Now we assume that . Let , where is odd and is an integer. Then we have and the -adic expansion of given by
[TABLE]
Since is odd, . We have
[TABLE]
It then follows that
[TABLE]
The desired conclusion on the first case then follows.
If , it follows from Lemma 3.1 that
[TABLE]
When , it is easy to see that
[TABLE]
Furthermore, one can similarly check that for and . Next, we assume that . Let , where is odd and is an integer. Then we have . Let the 2-adic expansion of be given by
[TABLE]
Since is odd, . Then
[TABLE]
As a result, we have
[TABLE]
This completes the proof. ∎
Lemma 3.3**.**
Let . Then we have the following.
If , then and
[TABLE] 2. 2.
If , then and
[TABLE]
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
When and , we have the following lemma on the defining sets and .
Lemma 3.4**.**
Let . Then we have the following.
If , then and
- •
,
- •
. 2. 2.
If , then and
- •
,
- •
.
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
B. The case:
Lemma 3.5**.**
Let . Then we have the following.
If , then and
[TABLE] 2. 2.
If , then and
[TABLE]
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
Lemma 3.6**.**
Let . Then we have the following.
If , then and
[TABLE] 2. 2.
If , then and
[TABLE]
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
Lemma 3.7**.**
Let . Then we have the following.
If , then and
- •
,
- •
. 2. 2.
If , then and
- •
,
- •
.
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
C. The case:
Lemma 3.8**.**
Let . Then we have the following.
If , then and
[TABLE] 2. 2.
If , then and
[TABLE]
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
Lemma 3.9**.**
Let . Then we have the following.
If , then and
[TABLE] 2. 2.
If , then and
[TABLE]
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
Lemma 3.10**.**
Let . Then we have the following.
If , then and
- •
,
- •
. 2. 2.
If , then and
- •
,
- •
.
Proof.
The proof is very similar to that of Lemma 3.2 and omitted here. ∎
4 Parameters of the codes and their related codes
In this section, we investigate the parameters of the codes and their related codes. We begin to consider the case that . When , the parameters of and are treated in the following theorem.
Theorem 4.1**.**
Let be an integer. Then and form a pair of odd-like duadic codes with parameters , where
[TABLE]
Proof.
It is known that and form a pair of duadic codes with length and dimension , so they have the same minimum distance .
We only prove the lower bounds on minimum distannce for as it is similar to prove the desired conclusion for . Denote . It follows from Lemma 3.2 that . Let be the integer satisfying . Write . It is deduced from Lemma 3.2 that defining set of with respect to contains the set . The lower bound on minimum distance of then follows from the BCH bound on the cyclic codes. This completes the proof. ∎
When , the following theorem provides lower bounds on minimum distances of the dual codes and .
Theorem 4.2**.**
Let be an integer. Then and form a pair of even-like duadic codes with parameters , where
[TABLE]
Proof.
It is easily seen that the defining sets of and with respect to are and , respectively. It then follows that is the even-weight subcode of and is the even-weight subcode of . The desired conclusion then follows from Theorem 4.1. ∎
When , the following theorem presents dimensions and lower bounds on minimum distances of the extended codes and .
Theorem 4.3**.**
Let be an integer. Then the extended codes and of and are self-dual and doubly-even, and they have parameters
[TABLE]
Proof.
It is well known that the extended codes of a pair of odd-like binary duadic codes are self-dual if the splitting corresponding to the pair of odd-like binary duadic codes is given by [7, Theorem 6.4.12]. As shown earlier, is a splitting of . Consequently, the extended codes and are self-dual. It then follows from [7, Theorem 6.5.1] that the Hamming weight of each codeword in and is divisible by . The remaining conclusions follow from Theorem 4.1. ∎
When or , we have the following theorem on parameters of the codes and and their related codes.
Theorem 4.4**.**
Let be an integer and suppose that or .
The odd-like duadic codes and have parameters
[TABLE] 2. 2.
The dual codes and form a pair of even-like duadic codes with parameters
[TABLE] 3. 3.
The extended codes and are self-dual and doubly-even, and they have parameters
[TABLE]
Proof.
It is very similar to those of Theorems 4.1, 4.2, 4.3 and omitted here. ∎
When , we have the following theorem on parameters of the codes and and their dual and extended codes. It can be similarly proved and we omit the details here.
Theorem 4.5**.**
Let be an integer.
When , we have the following.
The odd-like duadic codes and have parameters , where
[TABLE] 2. 2.
The codes and form a pair of even-like duadic codes with parameters , where
[TABLE] 3. 3.
The extended codes and are self-dual and doubly-even, and they have parameters
[TABLE]
When or , we have the following.
The odd-like duadic codes and have parameters
[TABLE] 2. 2.
The codes and form a pair of even-like duadic codes with parameters
[TABLE] 3. 3.
The extended codes and are self-dual and doubly-even, and they have parameters
[TABLE]
When , we have the following theorem on parameters of the codes and and their dual and extended codes. It can be similarly proved and we omit the details here.
Theorem 4.6**.**
Let be an integer.
When , we have the following.
The odd-like duadic codes and have parameters , where
[TABLE] 2. 2.
The codes and form a pair of even-like duadic codes with parameters , where
[TABLE] 3. 3.
The extended codes and are self-dual and doubly-even, and they have parameters
[TABLE]
When , we have the following.
The odd-like duadic codes and have parameters
[TABLE] 2. 2.
The codes and form a pair of even-like duadic codes with parameters
[TABLE] 3. 3.
The extended codes and are self-dual and doubly-even, and they have parameters
[TABLE]
When , we have the following.
The odd-like duadic codes and have parameters
[TABLE] 2. 2.
The codes and form a pair of even-like duadic codes with parameters
[TABLE] 3. 3.
The extended codes and are self-dual and doubly-even, and they have parameters
[TABLE]
It can be seen that the lower bounds on minimum distances of the odd-like and even-like duadic codes developed in this paper are very close to the square root bound.
Example 1**.**
Let and let be a generator of with . The parameters of and and their dual codes are documented in Table I.
5 Summary and concluding remarks
In this paper, we investigated the parameters of several families of binary duadic codes and their dual and extended codes, which partially solves one case (i.e. ) of the open problem proposed in [10]. The dimensions of these codes were determined explicitly and lower bounds on their minimum distances were presented.
The only known binary duadic codes with a good minimum distance or a good lower bound on their minimum distances are the following:
- •
Binary quadratic residue codes with parameters and their even-weight subcodes, where and is a prime [7, Section 6.6].
- •
The punctured binary Reed-Muller codes of order which has parameters , where is odd [1].
- •
Binary duadic codes and constructed in [12], and binary duadic codes and constructed in [10] (Lower bounds on their minimum distances are close to the square root bound).
The binary duadic codes constructed in this paper are not identical with the family of binary quadratic residue codes, as is composite for many odd . Some known binary duadic codes of length are listed in Table II, where means the punctured binary Reed-Muller code of length and order [1]. From Tables I and II, one can always find at least one subset such that the parameters of are different from those of punctured binary Reed-Muller codes and the binary duadic codes presented in [12] and [10].
There are only a few families of binary duadic codes whose minimum distances have a lower bound close to the square-root bound. It was shown that the lower bounds on minimum distances of the binary duadic codes constructed in this paper are close to the square root bound. Cyclic codes have many important applications in data storage and communication systems. However, it is not easy to construct cyclic codes with a large dimension and good minimum distance when the length is the product of some small primes [14]. It would be interesting to construct more families of binary duadic codes with good minimum distances.
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