5.1 BCH Codes—Part I
59
is that we always assume that the code length and the cardinality of the alphabet are
relatively prime, i.e., gcd(q, n) = 1, because this condition ensures that C has simple
roots. Additionally, throughout this book, we utilize the notation C [a] to denote the
cyclotomic coset containing a, where a is not necessarily the smallest number in
C [a] .
5.1.1 Construction I
In this subsection we construct new families of nonbinary CSS codes derived from
two distinct classical BCH codes, not necessarily dual-containing. To proceed further,
let us recall the so-called CSS construction given in Lemma 3.6.1.
Let q be a prime power. Let C 1 and C 2 denote two classical linear codes both
over the field F q with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q , respectively, such that
C 2 ⊂ C 1 . Then there exists an [[n, K = k 1 − k 2 , D]] q CSS quantum code where
D = min{wt (c) : c ∈ (C 1 \C 2 ) ∪ (C
⊥
2 \C
⊥
1 )}.
We start by showing Lemma 5.1.1.
Lemma 5.1.1 Let q ≥ 3 be a prime power and let n > q be an integer such that
gcd(q, n) = 1. Assume also that (q − 1) | n and m = ord n (q) ≥ 2 hold. Then each
of the q-ary cosets C [lr] has only one element, where r is given by n = r (q − 1), and
1 ≤ l ≤ q − 2 is an integer.
Proof Since rq = n + r holds, one has
(lr)q = l(n + r ) ≡ lr mod n;
hence
(lr)q
t
≡ lr mod n
for each 1 ≤ t ≤ m − 1, proving the lemma.
Lemma 5.1.1 is applied in the proof of Theorem 5.1.1.
Theorem 5.1.1 Assume that q > 3 is a prime power and n > q is an integer relatively prime with q. Assume also that (q − 1) | n and m = ord n (q) = 2 are true.
Then there exists a quantum code with parameters [[n, n − 4(r − 2) − 2, d ≥ r ]] q ,
where r is such that n = r (q − 1).
Proof Since n | (q
2
− 1) and because we consider only nonprimitive BCH codes,
it follows that r ≤ q. As gcd(q, n) = 1, one has r < q, so the inequalities (r −
2)q < n and r + (r − 2)q < n hold. We next show that all the q-ary cosets (modulo n of course) given by C [0] = {0}, C [1] = {1, q}, C [2] = {2, 2q}, C [3] =
{3, 3q}, . . . , C [r −2] = {r − 2, (r − 2)q}, C [r ] = {r }, C [r +1] = {r + 1, r + q},
59
is that we always assume that the code length and the cardinality of the alphabet are
relatively prime, i.e., gcd(q, n) = 1, because this condition ensures that C has simple
roots. Additionally, throughout this book, we utilize the notation C [a] to denote the
cyclotomic coset containing a, where a is not necessarily the smallest number in
C [a] .
5.1.1 Construction I
In this subsection we construct new families of nonbinary CSS codes derived from
two distinct classical BCH codes, not necessarily dual-containing. To proceed further,
let us recall the so-called CSS construction given in Lemma 3.6.1.
Let q be a prime power. Let C 1 and C 2 denote two classical linear codes both
over the field F q with parameters [n, k 1 , d 1 ] q and [n, k 2 , d 2 ] q , respectively, such that
C 2 ⊂ C 1 . Then there exists an [[n, K = k 1 − k 2 , D]] q CSS quantum code where
D = min{wt (c) : c ∈ (C 1 \C 2 ) ∪ (C
⊥
2 \C
⊥
1 )}.
We start by showing Lemma 5.1.1.
Lemma 5.1.1 Let q ≥ 3 be a prime power and let n > q be an integer such that
gcd(q, n) = 1. Assume also that (q − 1) | n and m = ord n (q) ≥ 2 hold. Then each
of the q-ary cosets C [lr] has only one element, where r is given by n = r (q − 1), and
1 ≤ l ≤ q − 2 is an integer.
Proof Since rq = n + r holds, one has
(lr)q = l(n + r ) ≡ lr mod n;
hence
(lr)q
t
≡ lr mod n
for each 1 ≤ t ≤ m − 1, proving the lemma.
Lemma 5.1.1 is applied in the proof of Theorem 5.1.1.
Theorem 5.1.1 Assume that q > 3 is a prime power and n > q is an integer relatively prime with q. Assume also that (q − 1) | n and m = ord n (q) = 2 are true.
Then there exists a quantum code with parameters [[n, n − 4(r − 2) − 2, d ≥ r ]] q ,
where r is such that n = r (q − 1).
Proof Since n | (q
2
− 1) and because we consider only nonprimitive BCH codes,
it follows that r ≤ q. As gcd(q, n) = 1, one has r < q, so the inequalities (r −
2)q < n and r + (r − 2)q < n hold. We next show that all the q-ary cosets (modulo n of course) given by C [0] = {0}, C [1] = {1, q}, C [2] = {2, 2q}, C [3] =
{3, 3q}, . . . , C [r −2] = {r − 2, (r − 2)q}, C [r ] = {r }, C [r +1] = {r + 1, r + q},