5.1 BCH Codes—Part I
61
also has minimum distance greater than or equal to r . Since the code C
⊥
2 is equivalent to C, C
⊥
2 also has minimum distance greater than or equal to r . Therefore, the
resulting CSS code has minimum distance greater than or equal to r .
Next we compute the dimension of the corresponding CSS code. We know that
the degree of the generator polynomial of a cyclic code is equal to the cardinality of
its defining set. Furthermore, the defining set Z 1 of C 1 has r − 1 disjoint cyclotomic
cosets. Moreover, all of them (except coset C 0 ) have two elements; hence Z 1 has
2(r − 2) + 1 elements. Therefore, C 1 has dimension k 1 = n − 2(r − 2) − 1. Similarly, C 2 has dimension k 2 = 2(r − 2) + 1. Thus the dimension of the corresponding
CSS code is n − 4(r − 2) − 2. Applying the CSS construction to the codes C 1 and
C 2 , one can get a quantum code with parameters [[n, n − 4(r − 2) − 2, d ≥ r ]] q .
The proof is complete.
We illustrate Theorem 5.1.1 by means of a graphical scheme:
C 1
C [0] C [1] C [2] . . . C [r −2]
C 2
C
C [r ] C [r +1] . . . C [2r −2] C [a 1 ] . . . C [a n ]
C 2
.
Observe that the union of the q-cosets C [0] , C [1] , . . . , C [r −2] is the defining set of
code C 1 ; the union of the cosets C [0] , C [1] , . . . , C [r −2] , C [a 1 ] , . . . , C [a n ] is the defining
set of C 2 , where C [a 1 ] , . . . , C [a n ] are the remaining cosets in order to complete the
set of all q-cosets. Finally, the union of the cosets C [r ] , C [r +1] , . . . , C [2r −2] is the
defining set of C.
As an immediate result we have
Corollary 5.1.1 Assume that all the hypotheses of Theorem 5.1.1 are valid. Then
there exists a quantum code with parameters [[n, n − 4(c − 2) − 2, d ≥ c]] q , where
2 ≤ c < r.
Proof Let C 1 be the cyclic code generated by the product of the minimal polynomials
M
(0)
(x)M
(1)
(x) · . . . · M
(c−3)
(x)M
(c−2)
(x),
and C 2 be the cyclic code generated by the product of the minimal polynomials
i
M
(i)
(x),
where i /
∈ {r, r + 1, . . . , r + c − 2} and i runs through the coset representatives mod
n. Proceeding similarly as in the proof of Theorem 5.1.1, the result follows.
61
also has minimum distance greater than or equal to r . Since the code C
⊥
2 is equivalent to C, C
⊥
2 also has minimum distance greater than or equal to r . Therefore, the
resulting CSS code has minimum distance greater than or equal to r .
Next we compute the dimension of the corresponding CSS code. We know that
the degree of the generator polynomial of a cyclic code is equal to the cardinality of
its defining set. Furthermore, the defining set Z 1 of C 1 has r − 1 disjoint cyclotomic
cosets. Moreover, all of them (except coset C 0 ) have two elements; hence Z 1 has
2(r − 2) + 1 elements. Therefore, C 1 has dimension k 1 = n − 2(r − 2) − 1. Similarly, C 2 has dimension k 2 = 2(r − 2) + 1. Thus the dimension of the corresponding
CSS code is n − 4(r − 2) − 2. Applying the CSS construction to the codes C 1 and
C 2 , one can get a quantum code with parameters [[n, n − 4(r − 2) − 2, d ≥ r ]] q .
The proof is complete.
We illustrate Theorem 5.1.1 by means of a graphical scheme:
C 1
C [0] C [1] C [2] . . . C [r −2]
C 2
C
C [r ] C [r +1] . . . C [2r −2] C [a 1 ] . . . C [a n ]
C 2
.
Observe that the union of the q-cosets C [0] , C [1] , . . . , C [r −2] is the defining set of
code C 1 ; the union of the cosets C [0] , C [1] , . . . , C [r −2] , C [a 1 ] , . . . , C [a n ] is the defining
set of C 2 , where C [a 1 ] , . . . , C [a n ] are the remaining cosets in order to complete the
set of all q-cosets. Finally, the union of the cosets C [r ] , C [r +1] , . . . , C [2r −2] is the
defining set of C.
As an immediate result we have
Corollary 5.1.1 Assume that all the hypotheses of Theorem 5.1.1 are valid. Then
there exists a quantum code with parameters [[n, n − 4(c − 2) − 2, d ≥ c]] q , where
2 ≤ c < r.
Proof Let C 1 be the cyclic code generated by the product of the minimal polynomials
M
(0)
(x)M
(1)
(x) · . . . · M
(c−3)
(x)M
(c−2)
(x),
and C 2 be the cyclic code generated by the product of the minimal polynomials
i
M
(i)
(x),
where i /
∈ {r, r + 1, . . . , r + c − 2} and i runs through the coset representatives mod
n. Proceeding similarly as in the proof of Theorem 5.1.1, the result follows.