Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights

It is shown that the residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$ (resp.\ $24k+8$) and minimum Lee weight $8k+4 \text{ or }8k+2$ (resp.\ $8k+8 \text{ or }8k+6$) is a binary extremal doubly even self-dual code for every positive integer $k$. A number of new self-dual $\mathbb{Z}_4$-codes of length $24$ and minimum Lee weight $10$ are constructed using the above characterization. These codes are Type I $\mathbb{Z}_4$-codes having the largest minimum Lee weight and the largest Euclidean weight among all Type I $\mathbb{Z}_4$-codes of that length. In addition, new extremal Type II $\mathbb{Z}_4$-codes of length $56$ are found.


Introduction
Self-dual codes are an important class of (linear) codes 1 for both theoretical and practical reasons. It is a fundamental problem to classify self-dual codes of modest length and determine the largest minimum weight among selfdual codes of that length. Among self-dual Z k -codes, self-dual Z 4 -codes have been widely studied because such codes have nice applications to unimodular lattices and (non-linear) binary codes, where Z k denotes the ring of integers modulo k and k is a positive integer with k ≥ 2. It is well known that the Nordstorm-Robinson, Kerdock and Preparata codes, which are some best known non-linear binary codes, can be constructed as the Gray images of some Z 4 -codes [8]. We emphasize that the Nordstorm-Robinson code can be constructed as the Gray image of the unique self-dual Z 4 -code of length 8 and minimum Lee weight 6. In this note, we pay attention to the minimum Lee weight from the viewpoint of a connection with the minimum distance of binary (non-linear) codes obtained as the Gray images. Rains [18] gave upper bounds on the minimum Lee weights d L (C) of self-dual Z 4 -codes C of length n. For even lengths n = 24k + ℓ, the upper bounds are given as d L (C) ≤ 8k + g(ℓ), where g(ℓ) is given by the following table: In this note, we study residue codes of self-dual Z 4 -codes having large minimum Lee weights. According to the above upper bounds, the minimum Lee weights of self-dual Z 4 -codes of lengths 24k and 24k + 8 are at most 8k + 4 and 8k + 8, respectively. It is shown that the residue code of a self-dual Z 4 -code of length 24k and minimum Lee weight 8k + 4 or 8k + 2 is a binary extremal doubly even self-dual code of length 24k for every positive integer k. It is also shown that the residue code of a self-dual Z 4code of length 24k + 8 and minimum Lee weight 8k + 8 or 8k + 6 is a binary extremal doubly even self-dual code of length 24k + 8. As a consequence, we show that the minimum Lee weight of a self-dual Z 4 -code of length 24k (resp. 24k + 8) is at most 8k (resp. 8k + 4) for every integer k ≥ 154 (resp. k ≥ 159). A number of new self-dual Z 4 -codes of length 24 and minimum Lee weight 10 are constructed using the above characterization. Some selfdual Z 4 -codes of length n and minimum Lee weight d L are also constructed for the cases (n, d L ) = (32, 14), (48,18), (56,18). Finally, we give a certain characterization of binary self-dual codes containing the residue codes of selfdual Z 4 -codes for some other lengths.
All computer calculations in this note were done by Magma [4].

Preliminaries
2.1 Self-dual Z 4 -codes Let Z 4 (= {0, 1, 2, 3}) denote the ring of integers modulo 4. A Z 4 -code C of length n is a Z 4 -submodule of Z n 4 . Two Z 4 -codes are equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. The dual code C ⊥ of C is defined The Hamming weight wt H (x), Lee weight wt L (x) and Euclidean weight wt E (x) of a codeword x of C are defined as n 1 (x) + n 2 (x) + n 3 (x), n 1 (x) + 2n 2 (x) + n 3 (x) and n 1 (x) + 4n 2 (x) + n 3 (x), respectively, where n i (x) is the number of components of x which are equal to i. The minimum Lee weight d L (C) (resp. minimum Euclidean weight d E (C)) of C is the smallest Lee (resp. Euclidean) weight among all non-zero codewords of C. The residue code C (1) of C is the binary code defined as C (1) = {c (mod 2) | c ∈ C}. If C is a self-dual Z 4 -code, then C (1) is doubly even [6].
The following characterization of the minimum Lee weights is useful.
A self-dual Z 4 -code which has the property that all Euclidean weights are divisible by eight, is called Type II. A self-dual Z 4 -code which is not Type II, is called Type I. A Type II Z 4 -code of length n exists if and only if n ≡ 0 (mod 8), while a Type I Z 4 -code exists for every length. It was shown in [3] that the minimum Euclidean weight d E (C) of a Type II Z 4 -code C of length n is bounded by d E (C) ≤ 8⌊ n 24 ⌋ + 8. A Type II Z 4 -code meeting this bound is called extremal. It was also shown in [19] that the minimum Euclidean weight d E (C) of a Type I Z 4 -code C of length n is bounded by d E (C) ≤ 8⌊ n 24 ⌋ + 8 if n ≡ 23 (mod 24), and d E (C) ≤ 8⌊ n 24 ⌋ + 12 if n ≡ 23 (mod 24).

Binary self-dual codes, covering radii and shadows
A binary code C is called self-dual if C = C ⊥ , where C ⊥ is the dual code of C under the standard inner product. Two binary self-dual codes C and C ′ are equivalent, denoted C ∼ = C ′ , if one can be obtained from the other by permuting the coordinates. A binary self-dual code C is doubly even if all codewords of C have weight divisible by four, and singly even if there is at least one codeword of weight congruent to 2 modulo 4. It is known that a binary self-dual code of length n exists if and only if n is even, and a binary doubly even self-dual code of length n exists if and only if n ≡ 0 (mod 8).
The minimum weight d(C) of a binary self-dual code C of length n is bounded by d(C) ≤ 4⌊ n 24 ⌋+6 if n ≡ 22 (mod 24), d ≤ 4⌊ n 24 ⌋+4 otherwise [14] and [16]. A binary self-dual code meeting the bound is called extremal.
The covering radius R(C) of a binary code C is the smallest integer R such that spheres of radius R around codewords of C cover the space Z n 2 . The covering radius is a basic and important geometric parameter of a code. A vector a of a coset U is called a coset leader of U if the weight of a is minimal in U and the weight of a coset U is defined as the weight of a coset leader. The covering radius is the same as the largest weight of all the coset leaders of the code (see [1]). The following bound is known as the Delsarte bound (see [1,Theorem 1]).
Let C be a binary singly even self-dual code and let C 0 denote the subcode of codewords having weight congruent to 0 modulo 4. Then C 0 is a subcode of codimension 1. The shadow S of C is defined to be C ⊥ 0 \ C. Shadows were introduced by Conway and Sloane [5], in order to provide restrictions on the weight enumerators of singly even self-dual codes. A binary self-dual code meeting the following bound is called s-extremal. [2]). Let C be a binary self-dual code of length n and let S be the shadow of C. Let d(C) and d(S) denote the minimum weights of C and S, respectively. Then d(S) ≤ n 2 + 4 − 2d(C), except in the case that n ≡ 22 (mod 24) and d(C) = 4⌊ n 24 ⌋ + 6, where d(S) = n 2 + 8 − 2d(C). We end this section by proposing the following lemma, which is obtained from [13, Theorems 2.1 and 2.2].

Lemma 2.3 (Bachoc and Gaborit
Lemma 2.4. Let C be a binary self-orthogonal code of length n. (i) If n is even, then there is a binary self-dual code containing C.
(ii) If n ≡ 0 (mod 8) and C is doubly even which is not self-dual, then there is a binary doubly even self-dual code containing C, and there is a binary singly even self-dual code containing C.
3 Characterization of the residue codes for lengths 24k and 24k + 8

Length 24k
As described in Section 1, the minimum Lee weight of a self-dual Z 4 -code of length 24k is at most 8k + 4. In this subsection, we consider self-dual Z 4 -codes of length 24k and minimum Lee weight 8k + 4 or 8k + 2.
Theorem 3.1. Let C be a self-dual Z 4 -code of length 24k. Suppose that the minimum Lee weight of C is 8k + 4 or 8k + 2. Then C (1) is a binary extremal doubly even self-dual code of length 24k.
Recently, the nonexistence of a self-dual Z 4 -code of length 36 and minimum Lee weight 16 has been shown in [10]. This result can be directly obtained by the bound in [18], which is given in Section 1, however, the approach in [10] can be generalized to the following alternative proof of the above theorem. Suppose that C (1) is not self-dual. Since C (1) is doubly even, by Lemma 2.4, there is a binary singly even self-dual code C satisfying that where C 0 denotes the doubly even subcode of C. By Lemma 2.1, C (1) ⊥ has minimum weight at least 4k + 1. By [16,Theorem 5], C has minimum weight 4k + 2. By Lemma 2.3, the minimum weight of the shadow of a binary singly even self-dual [24k, 12k, 4k + 2] code is at most 4k, which is a contradiction. Hence, C (1) is self-dual, that is, C (1) is extremal. This completes the alternative proof.
Remark 3.3. For lengths up to 24, optimal self-dual Z 4 -codes with respect to the minimum Hamming and Lee weights were widely studied in [17]. At length 24, the above theorem follows from [17, Theorem 2 and Corollary 5].
For length 24k, the only known binary extremal doubly even self-dual codes are the extended Golay code G 24 and the extended quadratic residue code QR 48 of length 48. The existence of a binary extremal doubly even selfdual code of length 72 is a long-standing open question. In addition, there is no binary extremal doubly even self-dual code of length 24k for k ≥ 154 [21]. Hence, we immediately have the following: Corollary 3.4. The minimum Lee weight of a self-dual Z 4 -code of length 24k is at most 8k for every integer k ≥ 154.

Length 24k + 8
As described in Section 1, the minimum Lee weight of a self-dual Z 4 -code of length 24k + 8 is at most 8k + 8. In this subsection, we consider self-dual Z 4 -codes of length 24k + 8 and minimum Lee weight 8k + 8 or 8k + 6.
Theorem 3.5. Let C be a self-dual Z 4 -code of length 24k + 8. Suppose that the minimum Lee weight of C is 8k + 8 or 8k + 6. Then C (1) is a binary extremal doubly even self-dual code of length 24k + 8.
Proof. Suppose that C (1) is not self-dual. Since C (1) is doubly even, by Lemma 2.4, there is a binary singly even self-dual code C satisfying that where C 0 denotes the doubly even subcode of C. By Lemma 2.1, C (1) ⊥ has minimum weight at least 4k + 3. Hence, C has minimum weight 4k + 4. By Lemma 2.3, the minimum weight of the shadow of a binary singly even self-dual [24k + 8, 12k + 4, 4k + 4] code is at most 4k, which is a contradiction. (ii) The above theorem can be proved by a similar argument to the proof of Theorem 3.1.
It is known that there is a binary extremal doubly even self-dual code of length 24k + 8 for k ≤ 4. In addition, since there is no binary extremal doubly even self-dual code of length 24k + 8 for k ≥ 159 [21], we immediately have the following: Corollary 3.8. The minimum Lee weight of a self-dual Z 4 -code of length 24k + 8 is at most 8k + 4 for every integer k ≥ 159.

Self-dual Z -codes having large minimum Lee weights
By using the characterizations of the residue codes, which are given in the previous section, a number of self-dual Z 4 -codes having large minimum Lee weights are constructed in this section.

Double circulant and four-negacirculant codes
Throughout this note, let A T denote the transpose of a matrix A and let I k denote the identity matrix of order k. An n × n matrix is circulant and negacirculant if it has the following form: r 0 r 1 · · · r n−2 r n−1 cr n−1 r 0 · · · r n−3 r n−2 cr n−2 cr n−1 . . . (1) is called a bordered double circulant Z 4 -code of length 2n, where R is an (n − 1) × (n − 1) circulant matrix and α, β, γ ∈ Z 4 . A Z 4 -code with generator matrix of the form: is called a four-negacirculant Z 4 -code of length 4n, where A and B are n × n negacirculant matrices. By considering bordered double circulant codes and four-negacirculant codes, we found self-dual Z 4 -codes of length 24k and minimum Lee weight 8k +2 (k = 1, 2) and self-dual Z 4 -codes of length 32 and minimum Lee weight 14. These codes were found under the condition that the residue codes are binary extremal doubly even self-dual codes, by Theorems 3.1 and 3.5. Selfdual Z 4 -codes of length 56 and minimum Lee weight 18 were also found.
For bordered double circulant codes, the first rows of R and (α, β, γ) in (1) are listed in Table 1. For four-negacirculant codes, the first rows of A and B in (2) are listed in Table 2. The minimum Lee weights d L determined by Magma are also listed. The 5th column in both tables indicates the Type of the code.
Hence, wt E (x) = 12. Therefore, C is a Type I Z 4 -code having minimum Euclidean weight 12.
We use the following method in order to verify that given two Z 4 -codes are inequivalent (see [7]). Let C be a self-dual Z 4 -code of length n. Let M t = (m ij ) be the A t × n matrix with rows composed of the codewords x with wt H (x) = t in C, where A t denotes the number of such codewords. For an integer k (1 ≤ k ≤ n), let n t (j 1 , . . . , j k ) be the number of r (1 ≤ r ≤ A t ) such that all m rj 1 , . . . , m rj k are nonzero for 1 ≤ j 1 < . . . < j k ≤ n. We consider the set S t,k = {n t (j 1 , . . . , j k ) | for any distinct k columns j 1 , . . . , j k }.
In [7], the authors claimed that there are two inequivalent bordered double circulant Type I Z 4 -codes of length 24 and minimum Lee weight 10. Unfortunately, this is not true. In fact, the number of such codes should be three not two. The codes D 24,i (i = 1, 2, 3) given in Table 1 are bordered double circulant Type I Z 4 -codes of length 24 and minimum Lee weight 10. In Table 3, we list S k = (max(S 9,k ), min(S 9,k ), #S 9,k ) (k = 1, 2, 3, 4) for the codes. This table shows that the three codes D 24,1 , D 24,2 , D 24,3 are inequivalent.  self-dual Z 4 -codes C with C (1) = C, and an explicit method for construction of these 2 k(k+1) 2 self-dual Z 4 -codes C with C (1) = C was given in [15,Section 3]. In our case, there are 2 78 self-dual Z 4 -codes C with C (1) = G 24 , and it seems infeasible to find all such codes. Using the above method, we tried to construct many self-dual Z 4 -codes. Then we stopped our search after we found 57 self-dual Z 4 -codes having minimum Lee weight 10 satisfying that the 57 codes and the three codes in Table 3 have distinct S 9,k (k = 1, 2, 3, 4). Hence, we have the following proposition. We denote the new codes by C 24,i (i = 1, 2, . . . , 57). In Figure 1, we list generator matrices for C 24,i , where we consider generator matrices in standard form ( I 12 , M i ) and only 12 rows in M i are listed, to save space.
According to the table, the largest minimum Lee weight for length 32 is 14. The code D 32 in Table 2 is a Type II Z 4 -code of length 32 and minimum Lee weight 14, which gives an explicit example of such codes. In addition, the code C 32 in Table 2 is a Type II Z 4 -code of length 32 and minimum Lee weight 14. We verified by Magma that C (1) 32 It is unknown whether the three codes are equivalent or not. There are five inequivalent binary extremal doubly even self-dual codes of length 32, two of which are QR 32 and RM(2, 5) (see [20,Table IV]). It is worthwhile to determine whether there is a self-dual Z 4 -code C having minimum Lee weight 14 with C (1) ∼ = C for each C of the remaining three codes.
The extended lifted quadratic residue Z 4 -code QR 48 of length 48 is a selfdual Z 4 -code having minimum Lee weight 18, which is an extremal Type II Z 4 -code. This is the only known self-dual Z 4 -code of length 48 and minimum Lee weight at least 18. Of course, QR (1) 48 is QR 48 . According to the table in [11], the largest minimum Lee weight among bordered double circulant self-dual Z 4 -codes of length 48 is 18. The code D 48 in Table 1 gives an explicit example of such codes. It is unknown whether D 48 is equivalent to QR 48 or not.
At length 56, under the condition that the residue code is a binary extremal doubly even self-dual code, we tried to construct a self-dual Z 4 -code having minimum Lee weight 20 or 22, but our search failed to do this. In this process, however, we found extremal Type II Z 4 -codes. The code C 56 in Table 2 is a Type II Z 4 -code of length 56 and minimum Lee weight 18. Hence, C 56 is extremal. According to the table in [11], the largest minimum Lee weight among bordered double circulant self-dual Z 4 -codes of length 56 is 18. The codes D 56,1 and D 56,2 in Table 1 give explicit examples of such codes. We verified by Magma that D 56,2 has minimum Euclidean weight 20. Since D 56,1 is Type II, D 56,1 is extremal. We verified by Magma that C (1) 56 and D (1) 56,1 have automorphism groups of orders 28 and 54, respectively. This shows that C 56 and D 56,1 are inequivalent. An extremal Type II Z 4 -code of length 56 given in [9] has the residue code of dimension 14. Hence, we have the following: It is unknown whether there is a self-dual Z 4 -code having minimum Lee weight 20, 22 or not.
At length 80, the minimum Lee weight of the extended lifted quadratic residue Z 4 -code was determined in [12] as 26. It is unknown whether there is a self-dual Z 4 -code having minimum Lee weight 28, 30 or not.
Proof. Since all cases are similar, we only give the details for the case (α, β) = (6, 4). By Lemma 2.4, there is a binary self-dual code C satisfying that where C 0 denotes the doubly even subcode of C. By Lemma 2.1, C (1) ⊥ has minimum weight at least 4k + 2. Hence, C has minimum weight 4k + 2 or 4k + 4. Suppose that C has minimum weight 4k+4. By Lemma 2.3, the minimum weight of the shadow C ⊥ 0 \ C of C is at most 4k − 1, which contradicts the minimum weight of C (1) ⊥ . Now, suppose that C has minimum weight 4k + 2. The weight of every vector of the shadow C ⊥ 0 \ C is congruent to 3 modulo 4 [5]. Since C ⊥ 0 has minimum weight at least 4k+2, the shadow has minimum weight at least 4k + 3. By Lemma 2.3, the minimum weight of the shadow C ⊥ 0 \ C of C is at most 4k + 3. Hence, C is s-extremal.
The situations in the following proposition are slightly different to that in the above proposition. However, a similar argument to the proof of the above proposition establishes the following proposition, and their proofs are omitted.
Proposition 5.2. Let C be a self-dual Z 4 -code of length 24k+α and minimum Lee weight 8k + β. Let C be a binary self-dual code containing C (1) .
(ii) Suppose that (α, β) = (16,8). If C is singly even, then C is an sextremal self-dual code having minimum weight 4k + 4. If C is doubly even, then C is extremal.