MinT - Best Known (260−50, 260, s)-Nets in Base 4

Best Known (260−50, 260, s)-Nets in Base 4

(260−50, 260, 1548)-Net over F₄ — Constructive and digital

Digital (210, 260, 1548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 26, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (184, 234, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 78, 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]

(260−50, 260, 11035)-Net over F₄ — Digital

Digital (210, 260, 11035)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁶⁰, 11035, F₄, 50) (dual of [11035, 10775, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁶⁰, 16384, F₄, 50) (dual of [16384, 16124, 51]-code), using
  - an extension C_e(49) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,49], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 50 [i]

(260−50, 260, 6193370)-Net in Base 4 — Upper bound on s

There is no (210, 260, 6193371)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 432406 694076 119731 036063 364897 782678 318417 846399 661974 674527 402509 594992 465988 995662 337196 682627 226772 551834 549114 478813 958462 599091 429080 194592 811096 768506 > 4²⁶⁰ [i]