A Note on New Near-Extremal Type I <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></semantics></math></inline-formula>-Codes of Length 48

Abstract

The subject of this work is self-dual Type I Z4-codes of length 48. It is known that the usual upper bound for the minimum Euclidean weight for the self-dual Z4-codes cannot be achieved for this length and type of codes; i.e., extremal Type I Z4-codes of length 48 do not exist. We are therefore looking for near-extremal Type I Z4-codes of length 48. The only known Type I near-extremal Z4-code of length 48 was constructed in 2015 by M. Harada. We adapted a known pseudo-random search method and found at least two new near-extremal Type I Z4-codes of length 48. These codes are the first Type I near-extremal Z4-codes with a residue code of minimum weight 8.

Keywords

• Z 4 -code%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="equation"><inline-formula> <mml:math id="mm1117"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-code
• Euclidean weight
• self-dual code
• Z 4 -code%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> Type I <named-content content-type="equation"><inline-formula> <mml:math id="mm1118"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-code
• Z 4 -code%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> near-extremal <named-content content-type="equation"><inline-formula> <mml:math id="mm1119"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-code