MinT - Best Known (157, 157+42, s)-Nets in Base 4

Best Known (157, 157+42, s)-Nets in Base 4

(157, 157+42, 1056)-Net over F₄ — Constructive and digital

Digital (157, 199, 1056)-net over F₄, using

1 times m-reduction [i] based on digital (157, 200, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 50, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(157, 157+42, 4541)-Net over F₄ — Digital

Digital (157, 199, 4541)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁹, 4541, F₄, 42) (dual of [4541, 4342, 43]-code), using
- 427 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 19 times 0, 1, 33 times 0, 1, 52 times 0, 1, 74 times 0, 1, 99 times 0, 1, 122 times 0) [i] based on linear OA(4¹⁸⁷, 4102, F₄, 42) (dual of [4102, 3915, 43]-code), using
  - constructionÂ X applied to C_e(41) âŠ‚ C_e(40) [i] based on
    1. linear OA(4¹⁸⁷, 4096, F₄, 42) (dual of [4096, 3909, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    2. linear OA(4¹⁸¹, 4096, F₄, 41) (dual of [4096, 3915, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [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]

(157, 157+42, 1467578)-Net in Base 4 — Upper bound on s

There is no (157, 199, 1467579)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645564 060159 134282 679061 158096 074867 953723 331170 546667 619814 302208 528865 960924 736577 601723 179500 456491 113653 906897 062428 > 4¹⁹⁹ [i]