5.3 BCH Codes—Part III
101
From Lemmas 5.3.6 and 5.3.7, it is easy to see that C is an [n, n − 2m(q
2
− 1) −
1, d ≥ 2q
2
+ 2] q 2 code. From Lemma 5.3.8, C is Hermitian dual-containing. Applying the Hermitian construction, an [[n, n − 4m(q
2
− 1) − 2, d ≥ 2q
2
+ 2]] q quantum code can be constructed. The proof is complete.
Corollary 5.3.2 Let q ≥ 4 be a prime power and n=q
2m
− 1, where m = ord n (q
2
) ≥
3. Then there exist quantum codes with parameters
• [[n, n − 2mc − 2, d ≥ c + 2]] q , where 1 ≤ c < q
2
− 1;
• [[n, n − 2m(q
2
− 1) − 2, d ≥ q
2
+ 2]] q ;
• [[n, n − 2m(c − 1) − 2, d ≥ c + 2]] q , where q
2
+ 1 ≤ c ≤ 2q
2
− 2.
Exercise 5.3.2 Prove Corollary 5.3.2.
5.3.3 Construction III
The result given in the sequence is analogous to Lemma 5.3.5.
Lemma 5.3.9 Let q = 2 be a prime power and n = q
m
− 1, where m = ord n (q) ≥
3. If s =
m−1
i=0
q
i , then the q-coset C [s] has only one element.
Proof Left to exercise.
Exercise 5.3.3 Prove Lemma 5.3.9.
The following two results are analogous to Lemmas 5.3.6 and 5.3.7, respectively.
Lemma 5.3.10 Let q ≥ 3 be a prime power and n = q
m
− 1, where m = ord n (q) ≥
3. Let s =
m−1
i=0
q
i . Then the following are true:
(a) the q-cosets of the form C [s+i] are mutually disjoints, where 1 ≤ i ≤ q − 1;
(b) the q-cosets of the form C [s− j] are mutually disjoints, where 1 ≤ j ≤ q − 1;
(c) the q-cosets of the form C [s+i] are mutually disjoints to the q-cosets of the form
C [s− j] , where 1 ≤ i, j ≤ q − 1.
Lemma 5.3.11 Let q ≥ 4 be a prime power and n = q
m
− 1, where m = ord n (q) ≥
3. Let s =
m−1
i=0
q
i . Then the following hold:
(a) the cosets of the form C [s+i] , where 1 ≤ i ≤ q − 1, contain m elements;
(b) the cosets of the form C [s− j] , where 1 ≤ j ≤ q − 1, contain m elements.
Exercise 5.3.4 Show Lemmas 5.3.10 and 5.3.11.