MinT - Best Known (74−17, 74, s)-Nets in Base 4

Best Known (74−17, 74, s)-Nets in Base 4

(74−17, 74, 1032)-Net over F₄ — Constructive and digital

Digital (57, 74, 1032)-net over F₄, using

4² times duplication [i] based on digital (55, 72, 1032)-net over F₄, using
- trace code for nets [i] based on digital (1, 18, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(74−17, 74, 2051)-Net over F₄ — Digital

Digital (57, 74, 2051)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁷⁴, 2051, F₄, 2, 17) (dual of [(2051, 2), 4028, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4⁷⁴, 4102, F₄, 17) (dual of [4102, 4028, 18]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁷⁴, 4103, F₄, 17) (dual of [4103, 4029, 18]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(14) [i] based on
      1. linear OA(4⁷³, 4096, F₄, 17) (dual of [4096, 4023, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. linear OA(4⁶⁷, 4096, F₄, 15) (dual of [4096, 4029, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [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]

(74−17, 74, 391164)-Net in Base 4 — Upper bound on s

There is no (57, 74, 391165)-net in base 4, because

1 times m-reduction [i] would yield (57, 73, 391165)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 89 203972 803953 775625 818738 695476 910880 943045 > 4⁷³ [i]