The Mass Formula for Self-Orthogonal and Self-Dual Codes over a Non-Unitary Non-Commutative Ring

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mi>p</mi></msub><mo>=</mo><mfenced separators="" open="⟨" close="⟩"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>|</mo><mi>p</mi><mi>a</mi><mo>=</mo><mi>p</mi><mi>b</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><mo>,</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mi>b</mi><mo>,</mo><mi>a</mi><mi>b</mi><mo>=</mo><mi>a</mi><mo>,</mo><mi>b</mi><mi>a</mi><mo>=</mo><mi>b</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≠</mo><mi>b</mi></mrow></semantics></math></inline-formula> and <i>p</i> is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>p</mi></msub></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><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo></mrow></semantics></math></inline-formula> and 7, in short lengths.