MinT - Best Known (260−58, 260, s)-Nets in Base 4

Best Known (260−58, 260, s)-Nets in Base 4

(260−58, 260, 1539)-Net over F₄ — Constructive and digital

Digital (202, 260, 1539)-net over F₄, using

4² times duplication [i] based on digital (200, 258, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(260−58, 260, 4157)-Net over F₄ — Digital

Digital (202, 260, 4157)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁶⁰, 4157, F₄, 58) (dual of [4157, 3897, 59]-code), using
- 54 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 53 times 0) [i] based on linear OA(4²⁵⁹, 4102, F₄, 58) (dual of [4102, 3843, 59]-code), using
  - constructionÂ X applied to C_e(57) âŠ‚ C_e(56) [i] based on
    1. linear OA(4²⁵⁹, 4096, F₄, 58) (dual of [4096, 3837, 59]-code), using an extension C_e(57) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,57], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 58 [i]
    2. linear OA(4²⁵³, 4096, F₄, 57) (dual of [4096, 3843, 58]-code), using an extension C_e(56) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,56], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 57 [i]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(260−58, 260, 972230)-Net in Base 4 — Upper bound on s

There is no (202, 260, 972231)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 432405 731379 831909 516448 910550 704700 284914 462342 370428 204970 984387 315704 170204 785516 062866 085087 249164 359010 719370 883126 246469 786647 122327 694500 698848 223536 > 4²⁶⁰ [i]