MinT - Best Known (125, 125+35, s)-Nets in Base 4

Best Known (125, 125+35, s)-Nets in Base 4

(125, 125+35, 1048)-Net over F₄ — Constructive and digital

Digital (125, 160, 1048)-net over F₄, using

trace code for nets [i] based on digital (5, 40, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(125, 125+35, 3466)-Net over F₄ — Digital

Digital (125, 160, 3466)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶⁰, 3466, F₄, 35) (dual of [3466, 3306, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁶⁰, 4112, F₄, 35) (dual of [4112, 3952, 36]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(4¹⁵⁸, 4110, F₄, 35) (dual of [4110, 3952, 36]-code), using
    - constructionÂ X applied to C([0,17]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(4¹⁵⁷, 4097, F₄, 35) (dual of [4097, 3940, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
      2. linear OA(4¹⁴⁵, 4097, F₄, 33) (dual of [4097, 3952, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(125, 125+35, 1022939)-Net in Base 4 — Upper bound on s

There is no (125, 160, 1022940)-net in base 4, because

1 times m-reduction [i] would yield (125, 159, 1022940)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 534003 608481 865282 014192 152143 787117 552092 594990 212205 736306 219492 225350 156256 152493 303432 098975 > 4¹⁵⁹ [i]