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In this paper, we study mixed alphabet codes over two alphabets Z 2 , and its extension Z 2 [ u ] = Z 2 + u Z 2 , u 2 = 0. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes. [ABSTRACT FROM AUTHOR]

For any odd prime p, we provide in this paper two new classes of AMDS symbol-pair cyclic codes of length 4p, one of symbol-pair distance 7 and one of symbol-pair distance 8. This results in two families of AMDS symbol-pair cyclic codes that can be used directly in applications. [ABSTRACT FROM AUTHOR]

BCH codes as a subclass of constacyclic BCH codes have been widely studied, while the results on the parameters of BCH codes over finite fields are still very limited. In this paper, we investigate some q-ary BCH codes and λ -constacyclic BCH codes of length q m + 1 , where q is a prime power and ord (λ) ∣ q - 1 . We determine the dimensions of these codes with some large designed distances, and give good lower bounds on the minimum distance. The code examples presented in this paper indicate that these codes contain many distance-optimal codes and codes with best known parameters. [ABSTRACT FROM AUTHOR]

There are various metrics for researching error-correcting codes. Especially, high-density data storage system gives the existence of inconsistency for the reading and writing process. The symbol-pair metric is motivated for outputs that have overlapping pairs of symbols in a certain channel. The Rosenbloom–Tsfasman (RT) metric is introduced since there exists a problem that is related to transmission over several parallel communication channels with some channels not available for the transmission. In this paper, we determine the minimum symbol-pair weight and RT weight of repeated-root cyclic codes over R = F p m [ u ] / ⟨ u 4 ⟩ of length n = p k . For the determination, we explicitly present third torsional degree for all different types of cyclic codes over R of length n. [ABSTRACT FROM AUTHOR]

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length n p s over p m , where p is a prime number and p does not divide n , the problem has been solved completely for all n ≤ 6 and partially for n = 7 , 8. In this paper, we solve the problem for n = 9 and all primes p different from 3 and 1 9. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length 9 p s over p m . As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes. [ABSTRACT FROM AUTHOR]

Negacyclic BCH codes are an important subclass of negacyclic codes and have good parameters. Inspired by the recent work on cyclic codes published in Wu et al. (Finite Fields Appl 60:101581, 2019), the objective of this paper is to investigate the parameters of the narrow-sense negacyclic BCH codes of length n = q 2 m - 1 q + 1 over GF (q) , where q is an odd prime power. For 2 ≤ δ ≤ q m + 1 - q q + 1 + 2 , the dimension of the narrow-sense negacyclic BCH codes of length n with designed distance δ is determined. For 2 ≤ δ ≤ q 2 m - 1 + 1 q + 1 , a lower bound on the minimum distance of the dual codes of the narrow-sense negacyclic BCH codes of length n with designed distance δ is presented. Compared with cyclic codes, we obtain some negacyclic BCH codes with better parameters. [ABSTRACT FROM AUTHOR]

BCH codes are an interesting class of cyclic codes with good error-correcting capability and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Let F q be the finite field of size q and n = q m - 1 , where m is a positive integer. Let C (q , m , δ) be the primitive narrow-sense BCH codes of length n over F q with designed distance δ . Denote s = m - t , r = m mod s and λ = ⌊ t / s ⌋ . In this paper, we mainly investigate the dimensions and Bose distances of the codes C (q , m , δ) with designed distance of the following two types: δ = q t + h , ⌈ m 2 ⌉ ≤ t < m , 0 ≤ h < q s + ∑ i = 1 λ - 1 q r + i s ; δ = q t - h , ⌈ m 2 ⌉ < t < m , 0 ≤ h < (q - 1) ∑ i = 1 s q i . This extensively extends the results on Bose distance in Ding et al (IEEE Trans Inf Theory 61(5):2351–2356, 2015). Moreover, the parameters of the hulls of the BCH code C (q , m , q t) are studied in some cases. [ABSTRACT FROM AUTHOR]

Cyclic codes are an important subclass of linear codes, they not only have good algebraic structure, but also are easy to be encoded and decoded. At present, researchers have constructed many optimal ternary cyclic codes, but the study on quinary cyclic codes is less developed. In this paper, by analyzing the solutions of certain equations over F 5 m , we construct some optimal quinary cyclic codes with three zeros and with parameters [ 5 m - 1 , 5 m - 2 - 2 m , 4 ] , [ 5 m - 1 , 5 m - 2 - 3 m 2 , 4 ] . Moreover, the weight distributions of two classes of their duals are also provided. [ABSTRACT FROM AUTHOR]

As a particular subclass of cyclic codes, BCH codes have wide applications in storage devices, communication systems, consumer electronics and other fields. However, parameters of BCH codes are unknown in general. In this paper, we investigate parameters of BCH codes of length q m + 1 λ where λ ∣ q + 1 .Some new techniques are employed to study the coset leaders. For any odd prime power q and m = 4 , 8 , or m ≥ 12 and m ≡ 4 (mod 8) , the second, the third and the fourth largest coset leaders modulo q m + 1 are determined, and the dimensions of some BCH codes of length q m + 1 with large designed distances are given. For 1 < λ < q + 1 , the first few largest coset leaders and the coset leaders modulo q m + 1 λ in the range 1 to q ⌊ (m + 1) / 2 ⌋ λ are studied, and the dimensions of some BCH codes of length q m + 1 λ are given as well. The BCH codes presented in this paper are LCD codes and have a sharper lower bound on the minimum distance than the well-known BCH bound. [ABSTRACT FROM AUTHOR]

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when p = 2 , which gives an affirmative answer to Heng's conjecture. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper we give three constructions of cyclic self-orthogonal codes over Z 2 k , for k ≥ 3 , from Boolean functions on n variables. The first construction for each k, 3 ≤ k ≤ n , yields a self-orthogonal Z 2 k -code of length 2 n + 2 with all Euclidean weights divisible by 2 k + 1. In the remaining two constructions, for each even n and k ≥ 3 , we generate a self-orthogonal Z 2 k -code of length 2 n + 1. All Euclidean weights in the constructed code are divisible by 2 2 k - 1 or 2 k + 1 , depending on which of the two constructions is used. [ABSTRACT FROM AUTHOR]

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring R = Z4 + uZ4 + u²Z4 with u³ = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). We introduce distinct Gray maps and show that the Gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over Z4. It is also shown that the Gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over Z4. Moreover, the structure of λ-constacyclic codes of odd length n over the ring R is determined and give some suitable examples. [ABSTRACT FROM AUTHOR]

Cryptology is a part of mathematics as encryption and decryption. The purpose of encryption is to make information incomprehensible when it is in the hands of unauthorized people. The receiver can decrypt the message that encrypted by the sender with helping of the key. The important point is that the key cannot be decrypted by other people. One Time Pad method solves this problem. The key is used only once each encryption in this method. So, the key becomes harder to guess. If the key is solved by unauthorized people, the message cannot be solved. Because of with each decryption, many meaningful messages are obtained. Every cyclic shift in a cyclic code constructs a new key and in each encryption is used the new key. Many keys are generated thanks to cyclic codes. In this paper, we improve the new encryption scheme by using the cyclic codes with One Time Pad method. [ABSTRACT FROM AUTHOR]

This research paper discusses the construction of novel and better quantum codes from the direct product of t -copies of ring R (discussed in Section 2), using cyclic codes over R by employing the CSS construction technique. Here, we present an overview of the structure and essential properties of a ring R. Furthermore, we analyze the distance-preserving nature of the Gray map in Subsection 2.1. Here, we also investigate maximum distance separable (MDS) codes by using the concept of quantum singleton defect (QSD), which indicates the overall quality of codes. To demonstrate the practicality of our findings, we provide illustrative examples implemented using the Magma software. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the behaviour of cyclic codes over a finite ring, where the ring S = Tq[v, w]/⟨v² - α², w³ - β²w, vw - wv⟩ is constructed using a field Tq of order q, with q being an odd prime power, and nonzero elements α and β from Tq. With the help of Gray images of cyclic code over the defined ring S, we construct LCD codes. A condition is also established for cyclic codes that contain their duals. Additionally, by using the self-orthogonal condition of cyclic codes over S, we are able to construct quantum codes. [ABSTRACT FROM AUTHOR]

For an odd prime p and a positive integer r, let q = p r . The objective of this article is to study cyclic codes over the ring S = F q [ u , v , w ] / ⟨ u 3 - u , v 2 - v , w 2 - w , u v , v u , u w , w u , v w - w v ⟩ and to construct new and better quantum and LCD codes from them. We give the structure of cyclic codes over the ring S and obtain dual-containing codes over F q as the Gray images of dual-containing cyclic codes over S . Using these dual-containing codes, we obtain quantum codes and determine their parameters using the decomposition of cyclic codes over the ring S . We provide many new and better-than-existing quantum codes. We also give a method to obtain linear complementary dual (LCD) codes over S using the decomposition of cyclic codes over the ring S . We obtain some optimal and best-known linear codes as Gray images of LCD codes over S . [ABSTRACT FROM AUTHOR]

Let $ s \geq 1 $ be a fixed integer. In this paper, we focus on generating cyclic codes over the ring $ \mathcal{R}(\alpha_1, \alpha_2, \ldots, \alpha_s) $, where $ \alpha_i \in \mathbb{F}_q\backslash \{0\} $, $ 1 \leq i \leq s $, by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing $ \mathbb{F}_q $ reversibility. The aim to introduce the $ \mathbb{F}_q $ reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases $ (A, \; T, \; G, \; C) $.

We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F 4 . An infinite family of 0-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic structure of 2 r 8 s -additive codes, where r and s are nonnegative integers, 2 (respectively, 8 ) denotes the finite field of order 2 (respectively, 8). We first give the generator polynomials of additive cyclic codes over 8 and then the generator polynomials of additive cyclic codes over 2 8 is also given. In addition, we introduce a linear map W : 2 8 → 2 , and study its properties. What's more, the dual of additive cyclic codes over 2 8 are investigated as well. And we get that the duals of any additive cyclic codes over 2 8 are also additive cyclic codes. Finally, separable 2 8 -additive cyclic codes are investigated. [ABSTRACT FROM AUTHOR]

As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except F q 2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings G R (2 r , 2) , where r ≥ 2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over G R (2 r , 2) and the family of linear cyclic codes of length 2n over Z 2 r . This correspondence helps to obtain additive conjucyclic codes over G R (2 r , 2) via known linear cyclic codes over Z 2 r . We prove that the trace dual C Tr of an additive conjucyclic code C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z 2 r contained in additive conjucyclic codes over G R (2 r , 2) . Last but not least, we explicitly derive the generator matrices for these codes. [ABSTRACT FROM AUTHOR]

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring R = Z4 + uZ4 + u²Z4 with u3 = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). We introduce distinct Gray maps and show that the Gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over Z4. It is also shown that the Gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over Z4. Moreover, the structure of λ-constacyclic codes of odd length n over the ring R is determined and give some suitable examples. [ABSTRACT FROM AUTHOR]

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 2 m - 1 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 Liu et al (Finite Field Appl 91:102270, 2023). 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. [ABSTRACT FROM AUTHOR]

MDS constacyclic codes over finite fields are important in both theory and practice. In this paper, all [q + 1,2,q] and [q + 1,3,q − 1] MDS λ-constacyclic codes over GF(q) are characterized, three classes of [q + 1,4,q − 2] MDS λ-constacyclic codes over GF(q) are constructed, and four classes of [q + 1,k,q − k + 2] MDS λ-constacyclic codes over GF(q) for variable k are presented. Rationales for distinguishing the class of cyclic codes, the class of λ-constacyclic codes with λ≠ 1, and the class of non-constacyclic linear codes are given. Some applications of MDS codes in cryptography are also summarised. [ABSTRACT FROM AUTHOR]

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance 3 and a class of quantum codes with minimum distance 4 are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a theory for constructing cyclic codes of odd length over the ring R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = v 2 = u v = v u = 0 , which plays an important role in DNA computing. A direct link between the elements of R and the 6 4 codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over R. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over R with their minimum Hamming distance will be studied as well. [ABSTRACT FROM AUTHOR]

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of F q P Q , where P = F q [ v ] 〈 v 3 − α 2 2 v 〉 and Q = F q [ u , v ] 〈 u 2 − α 1 2 , v 3 − α 2 2 v 〉 are nonchain finite rings and α i is in F q / { 0 } for i ∈ { 1 , 2 } , where q = p m with m ≥ 1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes. [ABSTRACT FROM AUTHOR]

This paper studies cyclic and (1 + 2v)-constacyclic codes over the ring Z4[v]/hv ³ − 1i. By introducing three different Gray maps, we show that the Gray images of cyclic codes are quasi-cyclic codes over Z4 and that of (1 + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Moreover, we show that the Gray image of skew (1 + 2v)-constacyclic code is a quasi-cyclic code over Z4. [ABSTRACT FROM AUTHOR]

Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length p k over the Galois ring GR (p 2 , m) is the key to construct self-dual cyclic codes of length p k n over the integer residue class ring Z p 2 for any positive integer n satisfying gcd (p , n) = 1 . So far, existing literature has only determined the number of these self-dual cyclic codes (Des Codes Cryptogr 63:105–112, 2012). In this paper, we give an efficient construction for all distinct self-dual cyclic codes of length p k over GR (p 2 , m) by using column vectors of Kronecker products of matrices with specific types. On this basis, we further obtain an explicit expression for all these self-dual cyclic codes by using binomial coefficients. [ABSTRACT FROM AUTHOR]

Let p be a prime with 5 | (p - 1) . Let S be a set of all repeated-root cyclic codes C = ⟨ g (x) ⟩ , (x 5 - 1) | g (x) , of length 5p over a field field F p , whose Hamming distances are at most 7. In this paper, we present a method to find all maximum distance separable (MDS) symbol-pair codes in S. By this method we can easily obtain the results in Ma and Luo (Des Codes Cryptogr 90:121–137, 2022) and new MDS symbol-pair codes, so we remain two possible MDS symbol-pair codes for readers. [ABSTRACT FROM AUTHOR]

In this paper, we give a general form of ideals over R [ X ] / 〈 f (X) p s 〉 when f is a monic irreducible polynomial in R[X] and R = F p m + u F p m + ... + u n F p m with un+ 1 = 0. For f(X) = X − 1 this gives a general form of cyclic codes over R. Furthermore, we apply the result to classify completely the cyclic codes of length 5ps over F p m + u F p m + u 2 F p m when p ≡ 2 (mod 5) or p ≡ 3 (mod 5) and m is odd. [ABSTRACT FROM AUTHOR]

Let F q be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of F q . The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R = F q [ u 1 , u 2 ] / 〈 u 1 2 − α 2 , u 2 2 − β 2 , u 1 u 2 − u 2 u 1 〉 . We decompose the ring R by using orthogonal idempotents Δ 1 , Δ 2 , Δ 3 , and Δ 4 as R = Δ 1 R ⊕ Δ 2 R ⊕ Δ 3 R ⊕ Δ 4 R , and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes. [ABSTRACT FROM AUTHOR]

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length 3 p s over p m + u p m + u 2 p m . Then, we compute their separation parameters which was useful to compute the Hamming distance in [8]. [ABSTRACT FROM AUTHOR]

Let m = 8 k and α be a primitive element of the finite field G F (2 m) , where k ≥ 2 is an integer. In this paper, a class of binary cyclic codes C (u , v) of length 2 m - 1 with two nonzeros α - u and α - v is studied, where (u , v) = (1 , (2 m - 1) / 17) . It turns out that C (u , v) has parameters [ 2 m - 1 , 2 m - m - 9 , 4 ] and is distance-optimal with respect to the Sphere Packing bound. The weight distribution of the dual of C (u , v) is also completely determined based on some results on Gaussian periods. [ABSTRACT FROM AUTHOR]

As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented. [ABSTRACT FROM AUTHOR]

Hermitian dual-containing codes play an important role in constructing quantum codes, constructing Hermitian dual-containing codes with optimal parameters is an interesting topic. The objective of this paper is to construct several classes of quantum codes with optimal parameters. Using cyclic codes, three classes of optimal Hermitian dual-containing codes are constructed. By employing the Hermitian construction, three classes of quantum codes with optimal parameters are obtained. These codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

A length ml, index l quasi-cyclic code can be viewed as a cyclic code of length m over the field Fql via a basis of the extension Fql/Fq. This cyclic code is an additive cyclic code. In [C. Güneri, F. Ozdemir, P. Sol'e, On the additive cyclic structure of quasi-cyclic codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize the (l,m) values for one-generator quasi-cyclic codes for which it is impossible to have an Fql -linear image for any choice of the polynomial basis of Fql/Fq. But this characterization for some (l,m) values is very intricate. In this paper, by the use of this characterization, we give a more simple characterization. [ABSTRACT FROM AUTHOR]

In this paper, we classify completely the negacyclic codes of length 3 p s over p m + u p m + u 2 p m when u 3 = 0 by using six-cyclotomic codes. [ABSTRACT FROM AUTHOR]

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we determine the dual codes of cyclic codes of length p s over R 3 = F p m [ u ] 〈 u 3 〉 , where p is a prime number and u 3 = 0. Also, we improve and give correction of the results stated by B. Kim and J. Lee (2020) in [11]. Finally, we provide some examples of optimal and near-MDS cyclic codes of length p s over R 3 and compute dual of them. [ABSTRACT FROM AUTHOR]

BCH codes have been widely used in consumer devices, communication systems, and data storage systems. In this paper, we study the dimensions of four families of the BCH codes C (q , n , δ , 0) of length n = q m + 1 , where designed distances δ are given as follows. (1) δ = a q m + 1 q + 1 + 1 , where 1 ≤ a ≤ ⌊ q − 1 2 ⌋ and m is odd. (2) δ = a q m − 1 q − 1 +2, where 1 ≤ a ≤ ⌊ q − 1 2 ⌋. (3) a q m − 1 q − 1 + 2 < δ < a q m − 1 q − 1 + (q − 2 a − 1) q m − 1 2 + 3 , where m is odd and 0 ≤ a ≤ ⌊ q − 3 2 ⌋. (4) a q m − 1 q − 1 + 2 < δ < a q m − 1 q − 1 + q m 2 + 4 , where m is even and 0 ≤ a ≤ ⌊ q − 3 2 ⌋. The minimum distance of the first family of the BCH codes C (q , n , δ , 0) is determined. The Bose distances of the third and fourth families of the BCH codes C (q , n , δ , 0) are determined under some conditions. Many optimal linear codes are obtained from these BCH codes. [ABSTRACT FROM AUTHOR]