New Quantum Codes Constructed by Quantum Caps in PG(3,9) and PG(4,9)

In this paper, we present a computer-supported method of searching for quantum caps. By means of this method and relevant knowledge of combinatorics, many quantum caps in <inline-formula> <tex-math notation="LaTeX">$PG(3,9)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$PG(4,9)$ </tex-math></inline-formula> are constructively proven to exist. Then, according to the theorem that each quantum cap corresponds to a quantum error-correcting code with <inline-formula> <tex-math notation="LaTeX">$d=4$ </tex-math></inline-formula>, we obtain 278 quantum error-correcting codes. Most of these results break the GV bound, and a number of them are optimal quantum codes or have improved parameters.


I. INTRODUCTION
Compared to classic computing, quantum computing has overwhelming superiority in terms of operation and security. In 1994, Shor [1] proposed a quantum computer-based algorithm that can factor an integer in polynomial time, which is impossible for a classic computer. However, due to the quantum incoherence effect, quantum computing is more likely to produce errors. Therefore, quantum error correction is essential in quantum computing.
In 1995, Shor [2] formulated the theory of quantum error-correcting codes (QECCs) and presented an example of a quantum [ [9,1,3]]-code that could correct one error. Since then, many methods of constructing QEECs have been being proposed. In 1998, Calderbank et al. [3] built the relationships between classical linear codes over the field F 4 and 2-ary QECCs, by which abundant QEECs with excellent parameters were identified. In 2006, Ketkar et al. [4] proposed a nonbinary construction theorem that translated the problem of finding q-ary QECCs into the problem of determining Hermitian self-orthogonal linear code. For a more detailed introduction to the construction of QECCs, refer to [5]- [8].
For this paper, the cited construction is given as follows. Lemma 1 [4]: If C is a q 2 -ary linear code of length n, dimension k and dual distance d ⊥ , which is self-orthogonal with respect to the Hermitian inner product, then there exists a pure quantum error-correcting code with parameters [[n, n − 2k, d ⊥ ]] q .
The associate editor coordinating the review of this manuscript and approving it for publication was Faissal El Bouanani .
One central theme in quantum error-correction is the construction of QECCs with optimal parameters. A quantum code [[n, k, d]] is optimal if there is no [[n, k, d + 1]] code. For the purpose of finding optimal QECCs with d = 4, researchers focus on quantum caps in PG(r, q). In [11], [12], the notion of the quantum cap is introduced. The authors note that a quantum cap of size n in PG(r, q) is equivalent to the quantum error-correcting code with parameters [[n, n − 2 (r +1), 4]] q . Thus, many 2-ary QEECs of optimal parameters are constructed by quantum caps in PG(r, 4) (see [9]- [15]).
The case of nonbinary optimal quantum codes is much more complicated. In this paper, we use mainly the quantum caps in PG (3,9) and PG (4,9) to construct 3-ary quantum codes. Among the results, most break the Gilbert-Varshamov (GV) bound, which implies having good parameters, and some are optimal according to the quantum Hamming bound. Moreover, compared to the 3-ary quantum codes of d = 3 in [16], [17], our results are more systematic and involve larger code lengths, some of which even have high code rates. Therefore, in some situations of one error correction in a quantum system, the quantum codes we propose are currently the best coding schemes.
Proposition 1 (Quantum Gilbert-Varshamov Bound [18]): Suppose that n > k ≥ 2, d ≥ 2 and n ≡ k(mod 2). Then, there exists a pure quantum code [[n, k, d]] q provided that Proposition 2 (Quantum Hamming Bound [18], [19]): For any pure quantum code [[n, k, d] The rest of this paper is organized as follows: basic concepts related to the linear code and projective cap are recalled in Sect.II. In Sect.III, a computer-supported method of searching for quantum caps is provided. In Sect.IV, 75 quantum caps in PG (3,9) are found, and related QECCs are constructed. In Sect.V, combinatorial construction and block processing are used to identify 203 quantum caps in PG (4,9), and related QECCs are obtained. Finally, we analyze the optimality of the constructed quantum codes and present conclusions.

A. FUNDAMENTALS OF LINEAR CODES
Let F q be a finite field with q elements, and let F n q be the n-dimensional vector space over F q . A k-dimensional subspace C of F n q is called a q-ary linear [n, k] code and is denoted as C = [n, k] q . If the minimal Hamming distance of C is d, For x = (x 1 , · · · , x n ) and y = (y 1 , · · · , y n )∈F n q , the Euclidean inner product is defined as If x, y∈F n q 2 , their Hermitian inner product is defined as If C = [n, k] q , its Euclidean dual code C ⊥ is For C = [n, k] q 2 , its Hermitian dual code C ⊥ h is ) be a generator matrix of C and G † = (g q i,j ) T be the conjugate transpose of G; then, C is Hermitian self-orthogonal if and only if G · G † = 0. Definition 1: Let C = [n, k] q and G = (α 1 , · · · , α n ) be a generator matrix of C. If J = {j 1 , · · · , j s } ⊆ {1, · · · , n} = [n], ω(J ) = {ω j 1 , · · · , ω j s } is a subset of nonzero elements in F q , G(ω(J )) = (β 1 , · · · , β n ) with β j l = ω j l α j l for j l ∈ J and β i = α i for i ∈ [n]\J ; then, the code C with generator matrix G(ω(J )) is an equivalent code of C. J and ω(J ) are called the varied set and varied value, respectively.

B. FUNDAMENTALS OF THE PROJECTIVE CAP
Let PG(r, q) be the r-dimensional projective space over F q . An n-cap in PG(r, q) is a set of n points, no three of which are collinear. Two caps in PG(r, q) with no common points are called disjoint caps. For two K 1 and K 2 caps in PG(r, q), if K 1 is a subset of K 2 , K 1 is called a subcap of K 2 , and this relation is denoted as K 1 ⊂ K 2 . An n-cap is called complete if it is not contained in an (n + 1)-cap. The n-cap in PG(r, q) with the largest size is called the maximal cap.
If we write the n points of an n-cap K in PG(r − 1, q) as columns of a matrix, we obtain an r × n matrix G r,n such that any three columns of G r,n are linearly independent, and G r,n is called a representative matrix of K . For two different representative matrices G r,n and G r,n of K , there are some J ⊆ [n] and ω(J ) such that G r,n = G r,n (ω(J )).
If C r,n is generated by G r,n , then C r,n is called a cap code. Clearly, C r,n is an [n, r] code with dual distance 4. When C r,n is a Hermitian self-orthogonal code, G r,n is called a quantum cap. Therefore, according to Lemma 1, we have the following.
Lemma 2 [7], [14]: The following are equivalent: • An [n, k] q 2 linear code of dual distance 4, which is self-orthogonal with respect to the Hermitian form.
From two special quantum caps in PG(r − 1, 9), we can obtain the following.
Lemma 3 [11]: Let G r,n and G r,m be quantum caps in PG(r − 1, 9) and m < n.
• If G r,m is a submatrix of G r,n , then there is a quantum n − m cap.
• If G r,n and G r,m are disjoint quantum caps, then there is a quantum n + m cap.
Reference [14] gives two results related to the quantum cap. VOLUME 8, 2020 Lemma 4 [14]: If C= [n, k] 9 , then C is Hermitian self-orthogonal if and only if (x, x) h = 0 for all x ∈ C.
The authors did not give a complete proof of the following lemma; thus, we provide a supplementary explanation.
Lemma 5 [14]: Let C be an [n, k] 9 with dual distance d. If there exists at least one codeword of N (C) ⊥ having weight m, then we can obtain a Hermitian self-orthogonal code with parameters [m, ≤ k] 9 and dual distance d.
Proof: Let G be a generator matrix of C, and let v = Case 1: Assume the nonzero coordinates of v consist of only 1; then, the obtained Hermitian self-orthogonal code can be generated based on some columns of G. For more details, refer to the proof of Theorem 2 in [14].
Case 2: Assume the nonzero coordinates of v consist of 1 and 2 and set J = {j| v j = 2} ⊆ {i 1 , i 2 , · · · i m }. Next, choose J as the varied set and let ω(J ) = {w}, where w is a primitive element of F 9 . For the sets A, B ⊆ F 9 , it is easy to check that w · A = B and w · B = A. Therefore, for equivalent code C generated by G(ω(J )), there must exist a codeword v ∈ N (C ) ⊥ with nonzero ordinates of all 1. By case 1, we can get a Hermitian self-orthogonal code with parameters [m, ≤ k] 9 and dual distance d generated by some columns of G(ω(J )).
Combining the above Lemma and proof with caps, we can easily obtain the following corollary.
Corollary 1: Suppose G r,n is an n-cap in PG(r − 1, 9) and v is a codeword of N (C r,n ) ⊥ having weight m.
(1) If the nonzero coordinates of v consist of only 1, then there exists a quantum m-cap from G r,n in PG(r , 9), r ≤ r − 1.
It is noteworthy that the dimensions of all obtained quantum caps in PG (3,9) and PG (4,9) are unchanged.
First, we choose G 4,82 as the starting point. From G 4,82 , we can directly find a quantum 8-cap that is defined as: According to Lemma 3, there is also a quantum 74-cap. Then, after computation, the norm code N (C 4,82 ) and its Euclidean dual code N (C 4,82 ) ⊥ can be obtained. We find that when 75 ≤ t ≤ 81, there exists a codeword of weight t whose nonzero coordinates consist of 1 and 2 contained in N (C 4,82 ) ⊥ . Thus, by Corollary 1, there exists quantum caps of sizes 75, 76, 77, 78, 79, 80, and 81 from G 4,82 (ω(J )) with different varied sets. These seven quantum caps and corresponding varied sets are listed in table 1.
On the basis of the above results, the following theorem is true.
Theorem 1: Assume s ∈ {8, 9, · · · , 82}; then, there exists a quantum s-cap in PG (3,9) and a related pure [[s, s − 8, 4]] 3 quantum error-correcting code. (4,9) In this section, to obtain quantum caps in PG (4,9), two solutions are proposed. The first is to use quantum caps in PG (3,9) to combinatorially construct quantum caps in PG (4,9). Second, we find the corresponding matrix of the 212-cap, which is the largest size of known cap in PG (4,9). We then divide this matrix into 5 blocks and search for quantum caps.

V. QUANTUM CAPS IN PG
Theorem 2: Let G r,m and G r,n be quantum subcaps of the same large cap in PG(r − 1, q) with sizes m and n, respectively. Denote the cap code of G r,m by C r,m . If C r,m contains a word of weight m, then there exists a quantum (m + n)-cap in PG(r, q).
, where c is a codeword of C r,m having weight m. Clearly, any three columns of G r+1,m+n are linearly independent, so G r,m+n is still a cap. Then, since G r,m and G r,n are quantum caps, we have c · G † r,m = 0, G r,m · G † r,m =G r,n · G † r,n = 0, c · (c ) † = 0. Thus, G r+1,m+n · (G r+1,m+n ) † = 0. G r+1,m+n is a quantum cap of size m + n in PG(r, q).

B. BLOCK PROCESSING OF THE 212-CAP IN PG
Through the first conclusion of Corollary 1, we search for all possible quantum caps that these five blocks contain. The search results are listed as follows.
The former four blocks of G 5,212 include quantum subcaps of sizes 30, 36, 30, and 33, respectively. If we denote the union of the former four blocks as G 5,172 , we can conclude that a quantum 129-cap is contained in G 5,172 . By deleting the columns of the quantum 129 subcap from G 5,172 , a new 43-cap can be obtained and defined as G 5,43 , shown at the bottom of the next page.
For G 5,43 , it is easy to check that it contains quantum caps of sizes 15, 18, 21, 24, and 27. In combination with the disjoint quantum 129-cap, we can construct quantum subcaps of G 5,172 with sizes of 144, 147, 150, 153, and 156. Moreover, since the last block G 5,40 contains quantum caps of size t, 11 ≤ t ≤ 30, which are disjoint with all the quantum caps from G 5,172 , one can derive by pairwise combination that there exists quantum caps in PG (4,9) of size t, where t ∈ {11, 12, · · · , 15} ∪ {164, 165, · · · , 186}. Next, the above discussion indicates that G 5,172 contains a quantum 156 subcap. Therefore, by deleting the columns of the quantum 156-cap from G 5,172 , we can obtain a 16-cap. Denote this 16-cap by G 5,16 . Then, by combining G 5,16 with the last block of G 5,212 , an 56-cap can be constructed and defined as G 5,56 = (G 5,16 , G 5,40 ). Denote the generated code of G 5,56 by C 5,56 . We can then obtain the norm code N (C 5,56 ) and its Euclidean dual code N (C 5,56 ) ⊥ .
After computation, we find that when t ∈ {31, 32, · · · , 44} ∪{48}, there exists a codeword of weight t whose nonzero coordinates consist of only 1 contained in N (C 5,56 ) ⊥ . By Corollary 1, there exists quantum caps of sizes t from G 5,56 . In addition, when t ∈ {10} ∪ {45, 46, · · · , 56}\{48}, N (C 5,56 ) ⊥ includes the codeword of weight t whose nonzero coordinates consists of 1 and 2. Thus, we can also obtain quantum caps with sizes of the same values from G 5,56 (ω(J )), where the varied sets J are all different. The sizes of the quantum caps and their related varied sets are listed below.
Then, by combining these results with the disjoint quantum 156 subcap of G 5,172 , we can obtain quantum cap of size t, 187 ≤ t ≤ 212, and a solitary quantum 10-cap in PG (4,9).
Thus, by combining all the results in Sect.V, the following theorem can be obtained.

VI. PARAMETER ANALYSIS OF QUANTUM CODE
In this section, the constructed quantum codes are analyzed in detail. It is easy to check that when 10   Comparison of the QECCs in this paper and those in [22], [23].  3 is an optimal code. In addition, by comparing all constructed results with the quantum codes in [16], [22], [23], 248 quantum codes are found to be new and some have better parameters. Here, we list only the QECCs that have better parameters than those of the quantum codes in [16], [22], [23].

VII. CONCLUSION
From the known maximum caps in PG (3,9) and PG (4,9), we have found 278 quantum caps of different sizes, whose representative matrices can be directly obtained. Then, by use of these quantum caps, we also constructed 278 related quantum error-correcting codes with d = 4 in succession. According to the GV bound and quantum Hamming bound, most of the quantum codes are optimal.
Although our codes could only correct one error in the quantum system, compared to the known 3-ary quantum codes of d = 3, our codes have larger sizes and parts of them have higher rates. In other words, for the case of a high rate with only one error correction required, the quantum codes we constructed are the best coding schemes at present.