MinT - Best Known (130−37, 130, s)-Nets in Base 8

Best Known (130−37, 130, s)-Nets in Base 8

(130−37, 130, 1026)-Net over F₈ — Constructive and digital

Digital (93, 130, 1026)-net over F₈, using

trace code for nets [i] based on digital (28, 65, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(130−37, 130, 4106)-Net over F₈ — Digital

Digital (93, 130, 4106)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³⁰, 4106, F₈, 37) (dual of [4106, 3976, 38]-code), using
- constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [i] based on
  1. linear OA(8¹²⁹, 4097, F₈, 37) (dual of [4097, 3968, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
  2. linear OA(8¹²¹, 4097, F₈, 35) (dual of [4097, 3976, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
  3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]

(130−37, 130, 3200187)-Net in Base 8 — Upper bound on s

There is no (93, 130, 3200188)-net in base 8, because

1 times m-reduction [i] would yield (93, 129, 3200188)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 315 216982 615821 096609 566263 395784 722493 657180 657867 309709 259733 601590 699706 651465 141527 077765 545456 374664 502828 725831 > 8¹²⁹ [i]