MinT - Best Known (178−37, 178, s)-Nets in Base 4

Best Known (178−37, 178, s)-Nets in Base 4

(178−37, 178, 1056)-Net over F₄ — Constructive and digital

Digital (141, 178, 1056)-net over F₄, using

4² times duplication [i] based on digital (139, 176, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 44, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(178−37, 178, 4570)-Net over F₄ — Digital

Digital (141, 178, 4570)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁸, 4570, F₄, 37) (dual of [4570, 4392, 38]-code), using
- 453 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0, 1, 75 times 0, 1, 101 times 0, 1, 129 times 0) [i] based on linear OA(4¹⁶⁴, 4103, F₄, 37) (dual of [4103, 3939, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
    1. linear OA(4¹⁶³, 4096, F₄, 37) (dual of [4096, 3933, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(4¹⁵⁷, 4096, F₄, 35) (dual of [4096, 3939, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(4¹, 7, F₄, 1) (dual of [7, 6, 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]

(178−37, 178, 2095377)-Net in Base 4 — Upper bound on s

There is no (141, 178, 2095378)-net in base 4, because

1 times m-reduction [i] would yield (141, 177, 2095378)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36695 980523 687253 803842 270858 822978 061872 278571 933817 348652 650336 667260 169197 114194 018641 125309 166033 572200 > 4¹⁷⁷ [i]