Cyclotomic Trace Codes

• Main Authors: Dean Crnković, Andrea Švob, Vladimir D. Tonchev
• Format: Article
• Language: English
• Published: MDPI AG 2019-08-01
• Series: Algorithms
• Subjects: linear code; two-weight code; strongly regular graph; block design
• Online Access: https://www.mdpi.com/1999-4893/12/8/168

A generalization of Ding&#8217;s construction is proposed that employs as a defining set the collection of the <i>s</i>th powers (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>&#8805;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>) of all nonzero elements in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>&#8805;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs.