| 要約: | The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="fraktur">F</mi><mi>q</mi></msub><mi>P</mi><mi>Q</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><msub><mi mathvariant="fraktur">F</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>v</mi><mo>]</mo></mrow></mrow><mrow><mo>⟨</mo><msup><mi>v</mi><mn>3</mn></msup><mo>−</mo><msubsup><mi>α</mi><mn>2</mn><mn>2</mn></msubsup><mi>v</mi><mo>⟩</mo></mrow></mfrac></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mo>=</mo><mfrac><mrow><msub><mi mathvariant="fraktur">F</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>]</mo></mrow></mrow><mrow><mo>⟨</mo><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>α</mi><mn>1</mn><mn>2</mn></msubsup><mo>,</mo><msup><mi>v</mi><mn>3</mn></msup><mo>−</mo><msubsup><mi>α</mi><mn>2</mn><mn>2</mn></msubsup><mi>v</mi><mo>⟩</mo></mrow></mfrac></mrow></semantics></math></inline-formula> are nonchain finite rings and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>α</mi><mi>i</mi></msub></semantics></math></inline-formula> is in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="fraktur">F</mi><mi>q</mi></msub><mo>/</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><msup><mi>p</mi><mi>m</mi></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> is a positive integer and <i>p</i> is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring <i>P</i>, we obtain several optimal codes with the help of the Gray image of cyclic codes.
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