A Construction Method of Quantum Error-correcting Codes over ${\boldsymbol F_{{2^m}}}$
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摘要: 构造具有良好参数的量子码是量子纠错码研究的重要内容。该文利用有限非链环
$ R = {F_{{4^m}}} + v{F_{{4^m}}} $ 上的厄米特对偶包含常循环码来构造$ {2^m} $ 元量子码。定义了一种新的Gray 映射$ \phi $ ，能够将环$ R $ 上线性码$ C $ 的厄米特对偶包含性保持到$ \phi (C) $ 上。研究了环$ R $ 上常循环码是厄米特对偶包含码的条件。给出了一种构造$ {2^m} $ 元量子码的方法，并构造了一些新的4元和8元量子码。Abstract: Constructing quantum codes with good parameters is an important part of quantum error-correcting codes research. In this paper,$ {2^m} $ -ary quantum codes are derived through Hermitian dual-containing constacyclic codes over finite non-chain ring$ R = {F_{{4^m}}} + v{F_{{4^m}}} $ . A new Gray map$ \phi $ is defined, which is Hermitian dual-containing preserving from a linear code C over R to$ \phi (C) $ . The condition for constacyclic codes over R to be Hermitian dual-containing is studied. A method of constructing$ {2^m} $ -ary quantum codes is presented, and some new 4-ary and 8-ary quantum codes are obtained.-
Key words:
- Constacyclic codes /
- Quantum codes /
- Finite non-chain ring /
- Gray map
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表 1 码长为34的新的4元量子码
$ {g_1}(x) $ $ {g_2}(x) $ $ \phi (C) $ $ {[[n,k,d]]_4} $ $ 1{\omega ^3}1 $ $ (1{\omega ^2}{\omega ^3})(1{\omega ^3}{\omega ^3}) $ [34,28,5]16 [[34,22,$ \ge $5]]4 $ (1{\omega ^3}1)(1{\omega ^6}1) $ $ (1{\omega ^2}{\omega ^3})(1{\omega ^3}{\omega ^3}) $ [34,26,6]16 [[34,18,$ \ge $6]]4 $ (1{\omega ^3}1)(1{\omega ^6}1) $ $ (1{\omega ^3}{\omega ^3})(1{\omega ^{11}}{\omega ^3})(1{\omega ^{13}}{\omega ^3}) $ [34,24,7]16 [[34,14,$ \ge $7]]4 $ (1{\omega ^3}1)(1{\omega ^6}1)(1\omega 1) $ $ (1{\omega ^3}{\omega ^3})(1{\omega ^{11}}{\omega ^3})(1{\omega ^{13}}{\omega ^3}) $ [34,22,8]16 [[34,10,$ \ge $8]]4 $ (1{\omega ^3}1)(1{\omega ^6}1)(1\omega 1) $ $ (1{\omega ^3}{\omega ^3})(1{\omega ^{11}}{\omega ^3})(1{\omega ^{13}}{\omega ^3})(1{\omega ^6}{\omega ^3}) $ [34,20,9]16 [[34,6,$ \ge $9]]4 
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表 2 新的4元量子码
$ n $ $ \lambda $ $ {g_1}(x) $ $ {g_2}(x) $ $ \phi (C) $ $ {[[n,k,d]]_4} $ $ {[[n',k',d']]_4} $ 3 $ 1 + v + v{\omega ^3} $ $ 1{\omega ^5} $ $ 1\omega $ $ {[6,4,3]_{16}} $ $ {[[6,2,3]]_4} $ MDS 7 $ 1 + v + v{\omega ^3} $ $ 1011 $ $ 1{\omega ^9}0{\omega ^{12}} $ $ {[14,8,6]_{16}} $ $ {[[14,2, \ge 6]]_4} $ $ {[[14,0,4]]_4}{\text{ }}$[15] 11 $ 1 + v + v{\omega ^3} $ $ 1{\omega ^5}11{\omega ^{10}}1 $ $ 1{\omega ^8}{\omega ^6}{\omega ^9}{\omega ^7}1 $ $ {[22,12,7]_{16}} $ $ {[[22,2, \ge 7]]_4} $ ${[[24,0,6]]_4}$[17] 15 $ 1 $ $ (1\omega )(1{\omega ^2}) $ $ 1{\omega ^4} $ $ {[30,27,3]_{16}} $ $ {[[30,24, \ge 3]]_4} $ $ {[[31,21,3]]_4} $[17] 17 $ 1 $ $ 1{\omega ^3}1 $ $ 1{\omega ^6}1 $ $ {[34,30,4]_{16}} $ $ {[[34,26, \ge 4]]_4} $ $ {[[34,24,4]]_4} $[15] 19 $ 1 + v + v{\omega ^3} $ $ 1{\omega ^{10}}0{\omega ^{10}}{\omega ^{10}}{\omega ^5}{\omega ^5}{\omega ^5}1 $ $ 1{\omega ^7}0\omega {\omega ^{13}}{\omega ^5}{\omega ^2}{\omega ^{11}}{\omega ^8} $ $ {[38,20,11]_{16}} $ $ {[[38,2, \ge 11]]_4} $ $ {[[40,2,8]]_4} $[17] 45 $ 1 $ $ (1\omega )(100{\omega ^4}) $ $ (1{\omega ^2})(100{\omega ^5}) $ $ {[90,82,4]_{16}} $ $ {[[90,74, \ge 4]]_4} $ $ {[[90,66,4]]_4} $[17] 63 $ 1 + v + v{\omega ^3} $ $ 111{\omega ^5} $ $ 1\omega $ $ {[126,122,3]_{16}} $ $ {[[126,118, \ge 3]]_4} $ $ {[[127,113,3]]_4} $[17] 77 $ 1 $ $ 1{\omega ^5}11{\omega ^{10}}1 $ $ 1011 $ $ {[154,146,4]_{16}} $ $ {[[154,138, \ge 4]]_4} $ $ {[[154,128,4]]_4} $[17] 85 $ 1 $ $ (1{\omega ^2}{\omega ^3})(1{\omega ^4}{\omega ^6}) $ $ (1{\omega ^9}{\omega ^9})(1{\omega ^8}{\omega ^{12}}) $ $ {[170,162,4]_{16}} $ $ {[[170,154, \ge 4]]_4} $ $ {[[171,151,4]]_4} $[17] 91 $ 1 + v + v{\omega ^3} $ $ 1{\omega ^4}{\omega ^{13}}1 $ $ (1{\omega ^3}{\omega ^8}{\omega ^9})(1{\omega ^7}{\omega ^4}{\omega ^9}) $ $ {[182,173,5]_{16}} $ $ {[[182,164, \ge 5]]_4} $ $[[185,149,5]]_4$[17]
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表 3 新的8元量子码
$ n $ $ \lambda $ $ {g_1}(x) $ $ {g_2}(x) $ $ \phi (C) $ $ {[[n,k,d]]_8} $ $ {[[n',k',d']]_8} $ $ 5 $ $ 1 + v + v{\omega ^7} $ $ 1{\omega ^{42}}1 $ $ 1{\omega ^{56}}{\omega ^{28}} $ $ {[10,6,5]_{64}} $ $ {[[10,2,5]]_8} $ MDS $ 7 $ $ 1 + v + v{\omega ^{21}} $ $ 1{\omega ^9} $ $ 1{\omega ^3} $ $ {[14,12,3]_{64}} $ $ {[[14,10,3]]_8} $ MDS $ 7 $ $ 1 + v + v{\omega ^{21}} $ $ 1{\omega ^9} $ $ (1{\omega ^3})(1{\omega ^{12}}) $ $ {[14,11,4]_{64}} $ $ {[[14,8,4]]_8} $ MDS $ 21 $ $ 1 + v + v{\omega ^{21}} $ $ 1{\omega ^3} $ $ 1\omega $ $ {[42,40,3]_{64}} $ $ {[[42,38,3]]_8} $ MDS $ 35 $ $ 1 + v + v{\omega ^{21}} $ $ (1{\omega ^9})(1{\omega ^{57}}{\omega ^9}) $ $ 1{\omega ^6} $ $ {[70,66,4]_{64}} $ $ {[[70,62, \ge 4]]_8} $ $ {[[70,46,3]]_8} $[17] $ 39 $ $ 1 + v + v{\omega ^{21}} $ $ (1{\omega ^{47}}{\omega ^{42}})(1{\omega ^{31}}{\omega ^{21}}) $ $ (1{\omega ^{27}}{\omega ^{35}})(1{\omega ^{45}}{\omega ^{14}}) $ $ {[78,70,5]_{64}} $ $ {[[78,62, \ge 5]]_8} $ $ {[[78,46,5]]_8} $[17] $ 49 $ $ 1 + v + v{\omega ^7} $ $ 1{\omega ^9} $ $ (1{\omega ^{22}})(1{\omega ^{31}}) $ $ {[98,95,3]_{64}} $ $ {[[98,92, \ge 3]]_8} $ $ {[[99,89,3]]_8} $[17] $ 63 $ $ 1 $ $ (1\omega )(1{\omega ^2}) $ $ (1{\omega ^3})(1{\omega ^4}) $ $ {[126,122,4]_{64}} $ $ {[[126,118, \ge 4]]_8} $ $ {[[127,113,3]]_8} $[17] $ 65 $ $ 1 + v + v{\omega ^{21}} $ $ (1{\omega ^4}1)(1{\omega ^8}1) $ $ (1{\omega ^{52}}{\omega ^{21}})(1{\omega ^{19}}{\omega ^{21}}) $ $ {[130,122,5]_{64}} $ $ {[[130,114, \ge 5]]_8} $ $ {[[133,113,5]]_8} $[17] $ 73 $ $ 1 + v + v{\omega ^7} $ $ 1{\omega ^{36}}01 $ $ 10{\omega ^{50}}{\omega ^{21}} $ $ {[146,140,4]_{64}} $ $ {[[146,134, \ge 4]]_8} $ $ {[[147,122,4]]_8} $[17] $ 91 $ $ 1 + v + v{\omega ^7} $ $ (1{\omega ^9})(1{\omega ^{31}}{\omega ^{36}}) $ $ (1{\omega ^{52}})(1{\omega ^{44}}{\omega ^{50}})(1{\omega ^{61}}) $ $ {[182,175,5]_{64}} $ $ {[[182,168, \ge 5]]_8} $ $ {[[183,143,4]]_8} $[17]
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