Quantum code construction literature reading 1---Quantum codes derived from cyclic codes

Today I read Giuliano G. La Guardia's article on the use of cyclocosets to construct quantum codes, and record it.

main idea:

Looking for some special cyclotomic cosets, so that there are r consecutive numbers in a cyclotomic coset, you can use this cyclotomic coset as the definition set to get the generator polynomial of the cyclic code, and then get the cyclic code, if this cyclotomic coset If the set is asymmetric, a class of quantum codes with better parameters can be constructed.

The main theorem:

Th1. (The BCH Bound) When the cyclic code $C$ generator polynomial is $g(x)$ , if present $b \ge 0$ , $\delta \ge 1$ of $F_q$ the elements of a domain expansion $\alpha$ , st

$g(\alpha^b)=g(\alpha^{b+1})=\cdots=g(\alpha^{b+\delta-2})=0$

It is $g(x)$ a continuous $\delta -1$ one $\alpha$ in the power of the root, the code $C$ minimum distance $\ge \delta$ .

Th2. (CSS construction method) If there is a classic linear $[n,k,d]_q$ code $C$ st $C^\perp \subset C$ , there is a $[[n, 2k-n, \ geq d]] _ q$ quantum code (the original content is stabilizer code that is pure, because I don’t know much about the quantum code itself, so I probably understand it as a quantum code. Welcome to the comment area Pointing).

Th3. Provided $q \ geq 3$ is a power of prime numbers, $n \ geq m$ is a positive integer and satisfy $gcd(q,n)=1$ and $gcd(q^{a_i}-1,n)=1$ of $\forall i = 1,2,\cdots,r$ which $m = ord_n (q) \ geq r + 2$ and $1 \leq r,a_1,a_2,\cdots,a_r <m$ an integer, if $n|gcd(t2,t3,\cdots,t_r)$ all $j=2,\cdots,r$ set up, wherein $t_j=[(j-(j-1)q^{a_j})(q^{a_j}-1)^{-1}-(q^{a_1}-1)^{-1}]$ , the presence of a $[n, nm ^ {\ star}, d \ geq r + 2] _q$ cyclic code, wherein $m^{\star}$ that there is $r+1$ a consecutive number of $q$ cyclotomic cosets The number of elements.

Th4. Provided $q \ geq 3$ is a power of a prime number $n \ geq m$ is a prime number and satisfies $gcd(q,n)=1$ , wherein $m = ord_n (q) \ geq r + 2$ and $1 \leq r,a_1,a_2,\cdots,a_r <m$ integer, if $n|gcd(t2,t3,\cdots,t_r)$ all $j=2,\cdots,r$ set up, where $t_j=[(j-(j-1)q^{a_j})(q^{a_j}-1)^{-1}-(q^{a_1}-1)^{-1}]$ , and $a_1,a_2,\cdots,a_r$ is an integer and satisfying $1\leq a_1+a_2,\cdots+a_r < m$ , there is a $[n, nm ^ {\ star}, d \ geq r + 2] _q$ cyclic code, wherein $m^{\star}$ that there is $r+1$ a consecutive number of $q$ points circle accompany The number of elements in the set.

Th4 is actually an inference of Th3, and the conditions are different, but the parameters of the final code are the same.

Th5. If the hypotheses in Th3 or Th4 are both true, it $C$ is $C_x$ a cyclic code defined by the set as , which $C_x$ is $r+1$ a coset containing a continuous integer, if $C_x \neq C_{-x}$ there is a $[[n, n-2m ^ {\ star}, d \ geq r + 2]] _ q$ quantum code.

References: https://arxiv.org/pdf/1705.00239.pdf