MinT - Best Known (215−46, 215, s)-Nets in Base 4

Best Known (215−46, 215, s)-Nets in Base 4

(215−46, 215, 1056)-Net over F₄ — Constructive and digital

Digital (169, 215, 1056)-net over F₄, using

1 times m-reduction [i] based on digital (169, 216, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 54, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(215−46, 215, 4456)-Net over F₄ — Digital

Digital (169, 215, 4456)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁵, 4456, F₄, 46) (dual of [4456, 4241, 47]-code), using
- 344 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 37 times 0, 1, 61 times 0, 1, 89 times 0, 1, 113 times 0) [i] based on linear OA(4²⁰⁵, 4102, F₄, 46) (dual of [4102, 3897, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(44) [i] based on
    1. linear OA(4²⁰⁵, 4096, F₄, 46) (dual of [4096, 3891, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    2. linear OA(4¹⁹⁹, 4096, F₄, 45) (dual of [4096, 3897, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [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]

(215−46, 215, 1334384)-Net in Base 4 — Upper bound on s

There is no (169, 215, 1334385)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2772 685697 836131 583166 316720 644662 072217 615365 968456 096916 647052 751247 018312 795019 093997 634769 090398 767138 566005 173645 055434 996704 > 4²¹⁵ [i]