ISSN 2148-838X

J. Algebra Comb. Discrete Appl.
9(3) • 185–191

Received: 8 August 2021
Accepted: 24 November 2021

Journal of Algebra Combinatorics Discrete Structures and Applications

A database of quantum codes

Research Article

Nuh Aydin, Peihan Liu, Bryan Yoshino

Abstract: Quantum error correcting codes (QECC) is becoming an increasingly important branch of coding
theory. For classical block codes, a comprehensive database of best known codes exists which is
available online at [17]. The same database contains data on best known quantum codes as well, but
only for the binary field. There has been an increased interest in quantum codes over larger fields
with many papers reporting such codes in the literature. However, to the best of our knowledge,
there is no database of best known quantum codes for most fields. We established a new database
of QECC that includes codes over Fq2 for q ≤ 29. We also present several methods of constructing
quantum codes from classical codes based on the CSS construction. We have found dozens of new
quantum codes that improve the previously known parameters and also hundreds new quantum codes
that did not exist in the literature.

2010 MSC: 94B05, 94B60, 94B65

Keywords: Quantum error-correcting codes, CSS construction, Constacyclic codes, Polycyclic codes

1. Introduction and motivation

Compared to classical information theory, the field of quantum information theory is relatively
young. The idea of quantum error correcting codes was first introduced in [31] and [11]. A method
of constructing quantum error correcting codes (QECC) was given in [10]. Since then researchers have
investigated various methods of using classical error correcting codes to construct new QECCs. There
are many publications in the literature with new quantum codes. However, compared to the classical
block codes, the database of quantum codes is very limited. While the database ([17]) covers the finite
fields of orders 2, 3, 4, 5, 7, 8 and 9, it only presents a database of QECC for q = 2. There are static
tables of some quantum codes given at [9], based on the work [8]. In recent years, researchers have
conducted searches for good quantum codes, and some results are presented in the papers listed in the
bibliography. We notice that the online tables [9] may have been overlooked by some researchers. In
many cases QECCs over finite fields of characteristic greater than 2 are presented in the literature. Hence

Nuh Aydin (Corresponding Author), Bryan Yoshino; Department of Mathematics, Kenyon College, USA (email:
aydinn@kenyon.edu, yoshino1@kenyon.edu).
Peihan Liu; Department of Mathematics, University of Michigan, USA (email: paulliu@umich.edu).

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codetables.de
http://codetables.de/
https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html
https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html


N. Aydin et. al. / J. Algebra Comb. Discrete Appl. 9(3) (2022) 185–191

there is a need to have an interactive database of QECCs that contains best known quantum codes over
larger alphabets which is continually updated. We now have compiled such a database. It is available
at http://quantumcodes.info, and it is being updated continually. In addition to the creation of the
database, we have also devised and implemented some search algorithms to find new quantum codes
from classical codes using the CSS construction. The paper also describes and reports the results of these
searches. One of the search methods we consider in this paper is about deriving quantum codes from
polycyclic codes associated with trinomials, which is based on the CSS construction. This methods has
been used in a recent work [3]. Here, we expand this method to construct more quantum codes.

This paper is organized as follows. In section 2, we recall some basic concepts about quantum codes
and polycyclic codes. In section 3, we present some search algorithms along with the new quantum codes
that we have found. A new code means it either has a better minimum distance than existing best
known quantum code with the same length and dimension over the same alphabet or a code with these
parameters does not exist in literature. All computations are performed using the MAGMA software.

2. Preliminaries

Let C be the field of complex numbers and consider the Hilbert space (Cq)⊗n = Cq ⊗Cq ⊗···⊗Cq︸ ︷︷ ︸
n times

.

A q-ary quantum code of length n and dimension k is a qk-dimensional subspace of (Cq)⊗n. If the
minimum distance of a quantum code is d then we denote it by [[n, k, d]]q. A connection between classical
block codes and quantum codes was given in [10] which has been used extensively in the literature. We
include the relevant results here.

Theorem 2.1. [10] Suppose S̄ is an (n−k)-dimensional linear subspace of Ē (a 2n-dimensional binary
vector space) which is contained in its dual S̄⊥ (with respect to the inner product, and is such that there
are no vectors of weight ≤ d − 1 in S̄⊥ \ S̄. Then there is a quantum-error-correcting code mapping k
qubits to n qubits which can correct [(d− 1)/2] errors.

Theorem 2.2. [10] Given two codes [[n1, k1, d1]] and [[n2, k2, d2]] with k2 ≤ n1 we can construct an
[[n1 + n2 −k2, k1, d]] code, where d ≥ min{d1, d1 + d2 −k2}.

The method of constructing quantum codes this way is known as CSS construction. There is a large
number of papers in the literature that present new quantum codes obtained by the CSS construction.
Recently, we have used polycyclic codes for the same purpose [3].

Definition 2.3. A linear code C is said to be polycyclic with respect to v = (v0, v1, ..., vn−1) ∈ Fnq if for
any codeword (c0, c1, ..., cn−1) ∈ C, its right polycyclic shift, (0, c0, c1, . . . , cn−2) + cn−1(v0, v1, . . . , vn−1)
is also a codeword. Similarly, C is left polycyclic with respect to v = (v0, v1, ..., vn−1) ∈ Fnq if for any
codeword (c0, c1, ..., cn−1) ∈ C, its left polycyclic shift (c1, c2, . . . , cn−1, 0) + c0(v0, v1, . . . , vn−1) is also a
codeword. If C is both left and right polycyclic, then it is bi-polycyclic.

In this work, we only work with right polycyclic codes, which we simply refer to as polycyclic codes.
Under the usual identification of vectors with polynomials, each polycyclic code C of length n is associated
with a vector v of length n (or a polynomial v(x) of degree less than n). We call v (v(x)) an associate
vector (polynomial) of C, and we say that C is a polycyclic code associated with xn − v(x). Moreover,
polycyclic codes of length n associated with f(x) = xn − v(x) are ideals of the factor ring Fq[x]/〈f(x)〉.
Note that an associate polynomial of a polycyclic code may not be unique.

Polycyclic codes are a generalization of cyclic codes and its several important generalizations. The
following are some of the most important special cases of polycyclic codes:

• A right polycyclic code with respect to v = (1, 0, 0, ..., 0) is a cyclic code.
A left polycyclic code with respect to v = (0, 0, 0..., 1) is a cyclic code.

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N. Aydin et. al. / J. Algebra Comb. Discrete Appl. 9(3) (2022) 185–191

• A right polycyclic code with respect to v = (−1, 0, 0, ..., 0) is a negacyclic code. A left polycyclic
code with respect to v = (0, 0, 0...,−1) is a negacyclic code.

• A right polycyclic code with respect to v = (a, 0, 0, ..., 0) is a constacyclic code. A left polycyclic
code with respect to v = (0, 0, 0, ..., a−1) is a constacyclic code.

3. Quantum CSS codes from polycyclic codes

Many of the works in the literature with new QECCs use the CSS construction method given in [10]
or something related to it. In this method, self-dual, self-orthogonal and dual-containing linear codes are
used to construct quantum codes. The CSS construction requires two linear codes C1 and C2 such that
C⊥2 ⊆ C1. Hence, if C1 is a self-dual code, then we can construct a CSS quantum code using C1 alone
since C⊥1 ⊆ C1. If C1 is self-orthogonal, then we can construct a CSS quantum code with C⊥1 and C1
since C1 ⊆ C⊥1 . Similarly in the case when C1 is a dual-containing code.

3.1. CSS codes by two polycyclic codes associated with trinomials

By the definition of CSS construction, we need two codes such that one is contained in the dual of
the other one. Let C be a polycylic code generated by g(x). By ideal inclusion we know that

〈g(x)f(x)〉⊆ 〈g(x)〉

for any polynomial f(x)and we consider C⊥2 = 〈g(x)f(x)〉 ⊆ 〈g(x)〉 = C1. Here g(x) is a divisor of
a trinomial t(x) = xn − axi − b and it generates a polycyclic code associated with t(x). Then to find
subcodes of C1, we use the factorization of t(x) into irreducibles. For each divisor f(x) of t(x) we get a
subcode of C1. The following results from [3] are useful in formulating the search.

Lemma 3.1. [3] Fix n, i ∈ Z, and let a1, a2, b ∈ F∗q be nonzero with a1 6= a2. Then gcd(xn − a1xi −
b, xn −a2xi − b) = 1

Lemma 3.2. [3] Fix n, i ∈ Z, and let a, b1, b2 ∈ Fq, with b1 6= b2. Then gcd(xn−axi−b1, xn−axi−b2) = 1.

Theorem 3.3. [3] Let n and i be positive integers with i < n. For two distinct trinomials xn − axi − b
and xn − cxi −d, we have gcd(xn −axi − b, xn − cxi −d) is either 1 or a binomial of degree gcd(n, i).

Lemma 3.4. [3] For any xn−axi−b 6= xn−cxi−d ∈ F3[x], we have gcd(xn−axi−b, xn−cxi−d) = 1.

We present some examples of quantum codes with new parameters in Table 1 where t(x) refers to
the trinomial, g1(x) is the generator of C1, and h2(x) is the generator of C⊥2 , i.e. 〈g1(x)〉 = C1 and
〈g1(x)f(x)〉 = 〈h2(x)〉 = C⊥2 . The last column in the table refers to the literature where a comparable
code is presented. Every code in this table has a higher minimum distance than the comparable code
presented in the literature. In the tables below, the coefficients of a polynomial are listed in increasing
powers of x, with leading coefficient at the right. Also, for alphabet of size greater than 10, A denotes
10, B denotes 11, ..., H denotes 17. Hence the entry [79F 1 for g1 on the first row of Table 2 represents
the polynomial 7 + 9x + 15x2 + x3

Table 1: New quantum codes constructed from polycyclic codes associ-
ated with trinomials

[[n, k, d]]q2 t(x) g1(x) h2(x) Ref.

[[60, 36, 5]]32 x
60 − 2x6 − 2 [1121011011001] [1222021012001] [7]

[[81, 52, 5]]32 x
81 − x30 − 1 [200210101101001] [1021111212022001] [29]

[[80, 54, 4]]52 x
80 − x − 1 [4231342331] [101124230104302301] [6]

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[[96, 80, 4]]52 x
96 − x − 1 [310032201] [101110301] [2]

[[52, 28, 4]]112 x
52 − 2x2 − 2 [7263851] [84782120107151973361] [23]

[[11, 1, 5]]132 x
11 − x − 3 [8A0C31] [352721] [20]

[[7, 1, 4]]172 x
7 − x2 − 2 [BFC1] [99101] [20]

3.2. CSS codes by a polycyclic code associated with trinomials and a random
linear code

In the search above, C1 and C2 are both polycyclic codes associated with the same trinomial.
However, this is not required. Hence, we can find some f(x) with desired degrees, and then construct C2
by g(x)f(x), where C1 = 〈g(x)〉 and C2 = 〈g(x)f(x)〉, i.e., C2 is not neccessarily polycyclic.

Note that this method works particularly well for target search by which we mean, we fix the field q,
length n and dimension k, and find the best known minimum distance for this set of parameters. Next,
we randomly generate a number of polynomials f(x) of degree k and iterate through divisors g(x) of t(x),
where t(x) is a trinomial, so that the CSS code constructed by C1 = 〈g(x)〉 and C⊥2 = 〈g(x)f(x)〉 is in
the form [[n, k]]q2.

Table 2: New quantum codes constructed from a polycyclic code and a
random linear code

[[n, k, d]]q2 t g1 g2 Ref.

[[7, 1, 4]]172 x
17 − x − 1 [79F1] [36DD1] [20]

[[24, 18, 3]]172 x
24 − x − 1 [EG41] [A6872366CA28406AE407F1] [16]

[[36, 30, 3]]172 x
36 − x − 1 [3791] [94F9161A374963B0E663E35AA4A6EFDGG1] [16]

[[48, 36, 4]]172 x
48 − x − 3 [CD3BE1] [60942809G709FE411F4910DCC6A7B5BC1545D270681][16]

3.3. CSS construction by two polycyclic codes associated with multinomials

It is also possible to expand the search space by considering multinomials instead of trinomials. Even
though data suggests that binomials and trinomials have more divisors than other multinomials, we still
found some good codes using this search. Some good quantum code have been found using this method
in [3].

Table 3: New quantum codes constructed from multinomials

[[n, k, d]]q2 m(x) g1(x) g2(x) Ref.

[[36, 6, 5]]32 [2011110210112212022010120211000010112102000101111] [222101212100200021] [222101212100200021][26]
[[22, 2, 6]]52 [13343122443414240410122] [4223310221] [442143133401] [9]

4. Obtaining new codes from existing codes

Like classical codes, we can also obtain new quantum codes from existing ones using the standard
method of extending (E), shortening (S) or puncturing (P) a code or taking the direct sum (DS) of two
given codes. All of these standard constructions are available in the Magma software. We applied these
constructions on exiting coded and obtained many new codes that have better minimum distances than

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the codes with the same length and dimension presented in the literature. In many cases we started with
the codes in the tables [9].

Table 4: New quantum codes from existing codes

[[n, k, d]]q2 Existing Code(s) (Method) [[n
′, k′, d′]]q2 Ref.

[[41, 1, 10]]32 [[33, 1, 10]] (E) [[41, 1, 8]]32 [9]
[[45, 21, 7]]32 [[41, 21, 7]] (E) [[45, 21, 5]]32 [7]
[[65, 29, 6]]32 [[25, 7, 6]], [[40, 22, 6]] (DS) [[65, 29, 5]]32 [9]
[[119, 7, 13]]32 [[52, 3, 13]], [[67, 4, 13]] (DS) [[119, 7, 11]]32 [9]
[[120, 6, 14]]32 [[53, 3, 14]], [[67, 3, 14]] (DS) [[20, 6, 12]]32 [9]
[[120, 96, 6]]32 [[121, 96, 7]] (P) [[120, 96, 4]]32 [7]
[[34, 2, 8]]52 [[26, 2, 8]] (E) [[34, 2, 6]]52 [9]
[[38, 2, 8]]52 [[26, 2, 8]] (E) [[38, 2, 7]]52 [9]
[[80, 48, 5]]52 [[11, 1, 5]], [[69, 47, 6]] (DS) [[80, 48, 2]]52 [19]
[[80, 54, 7]]52 [[78, 54, 7]] (E) [[80, 54, 3]]52 [6]
[[80, 56, 5]]52 [[16, 4, 5]], [[64, 52, 5]] (DS) [[80, 56, 3]]52 [15]
[[88, 8, 12]]52 [[52, 8, 12]] (E) [[88, 8, 5]]52 [19]
[[88, 48, 7]]52 [[22, 2, 7]], [[66, 46, 7]] (DS) [[88, 48, 2]]52 [25]
[[99, 3, 11]]52 [[29, 1, 11]], [[70, 2, 11]] (DS) [[99, 3, 9]]52 [9]
[[27, 17, 5]]72 [[25, 17, 5]] (E) [[27, 17, 3]]72 [25]
[[49, 40, 5]]72 [[48, 40, 5]] (E) [[49, 40, 3]]72 [29]
[[60, 36, 5]]72 [[18, 8, 5]], [[42, 28, 5]] (DS) [[60, 36, 4]]72 [25]
[[55, 35, 6]]72 [[58, 35, 9]] (P) [[55, 35, 4]]72 [9]

5. Features of the database

We introduce an interactive, online database [4] which stores the parameters of best known [[n, k, d]]q2
QECCs over fields GF (q2) for q ≤ 29 where q is a power of a prime and n ≤ 200. Users of the database
can search for best known QECCs by fixing q, n and k or by fixing q, n and d. Also, users can search
for multiple QECCs at the same time by fixing q and searching over ranges for n and d. Every QECC
displayed in the search results has a reference which is a link to the paper in which the code was introduced
or an online table where it is listed. In addition, all MDS codes are marked with an asterisk.

To construct such a database, we surveyed recent literature on QECCs and kept a record of pa-
rameters of codes in the literature. The bibliography gives the complete list of papers that have been
examined.

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https://doi.org/10.1142/S0219749909005560
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https://doi.org/10.1142/S021798491350053X
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https://doi.org/10.28919/jmcs/5593
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https://doi.org/10.1007/s10773-020-04673-0

	Introduction and motivation
	Preliminaries
	Quantum CSS codes from polycyclic codes
	Obtaining new codes from existing codes
	Features of the database
	References