Chapter 5
Quantum Code Constructions
Quantum codes are fundamental to the error protection in quantum computers.
Several families of good or optimal quantum codes were constructed by several
researches over time [4, 17, 25, 27, 28, 30, 39, 45, 46, 54, 55, 62, 72–74, 77–81,
89, 91, 105, 106, 111, 138, 147, 148, 162, 165]. The Calderbank–Shor–Steane
(CSS) construction is a remarkable technique to construct quantum codes derived
from classical ones. As was said previously, such technique has been applied by
a great number of quantum coding researchers in order to derive efficient quantum codes. With the possible advent of efficient quantum computers, it is extremely
important to investigate how to obtain families of efficient quantum codes. Based on
these facts, we present here some constructions of quantum codes derived from several classes of classical codes by means of the CSS construction. The classical codes
utilized here are the well-known Bose–Chaudhuri–Hocquenghem (BCH) (Sects. 5.1,
5.2, 5.3), algebraic geometry codes (Sect. 5.4) and quantum synchronizable codes
(Sect. 5.5).
We invite the reader to start our journey through the quantum code constructions.
The first families of quantum codes exhibited in Sect. 5.1 are obtained by applying the
CSS construction to suitable families of (classical) Bose–Chaudhuri–Hocquenghem
codes constructed carefully in order to attain codes which have, at the same time,
large dimension and minimum distance.
5.1 BCH Codes—Part I
In this subsection, we present five quantum code constructions generating several
families of nonbinary quantum BCH (see [22, 23, 63] for the first papers which originated such class of cyclic codes) with good parameters. The first two ones are based
on the CSS construction derived from two nonprimitive BCH codes. The third construction is based on Steane’s enlargement of nonbinary CSS codes applied to suitable
sub-families of nonprimitive non-narrow-sense BCH codes. The fourth construction
is obtained from suitable sub-families of Hermitian dual-containing nonprimitive
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_5
57
Quantum Code Constructions
Quantum codes are fundamental to the error protection in quantum computers.
Several families of good or optimal quantum codes were constructed by several
researches over time [4, 17, 25, 27, 28, 30, 39, 45, 46, 54, 55, 62, 72–74, 77–81,
89, 91, 105, 106, 111, 138, 147, 148, 162, 165]. The Calderbank–Shor–Steane
(CSS) construction is a remarkable technique to construct quantum codes derived
from classical ones. As was said previously, such technique has been applied by
a great number of quantum coding researchers in order to derive efficient quantum codes. With the possible advent of efficient quantum computers, it is extremely
important to investigate how to obtain families of efficient quantum codes. Based on
these facts, we present here some constructions of quantum codes derived from several classes of classical codes by means of the CSS construction. The classical codes
utilized here are the well-known Bose–Chaudhuri–Hocquenghem (BCH) (Sects. 5.1,
5.2, 5.3), algebraic geometry codes (Sect. 5.4) and quantum synchronizable codes
(Sect. 5.5).
We invite the reader to start our journey through the quantum code constructions.
The first families of quantum codes exhibited in Sect. 5.1 are obtained by applying the
CSS construction to suitable families of (classical) Bose–Chaudhuri–Hocquenghem
codes constructed carefully in order to attain codes which have, at the same time,
large dimension and minimum distance.
5.1 BCH Codes—Part I
In this subsection, we present five quantum code constructions generating several
families of nonbinary quantum BCH (see [22, 23, 63] for the first papers which originated such class of cyclic codes) with good parameters. The first two ones are based
on the CSS construction derived from two nonprimitive BCH codes. The third construction is based on Steane’s enlargement of nonbinary CSS codes applied to suitable
sub-families of nonprimitive non-narrow-sense BCH codes. The fourth construction
is obtained from suitable sub-families of Hermitian dual-containing nonprimitive
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_5
57