MinT - Best Known (129, 162, s)-Nets in Base 8

Best Known (129, 162, s)-Nets in Base 8

(129, 162, 2076)-Net over F₈ — Constructive and digital

Digital (129, 162, 2076)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (108, 141, 2048)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴¹, 2048, F₈, 33, 33) (dual of [(2048, 33), 67443, 34]-NRT-code), using
    - OOA 16-folding and stacking with additional row [i] based on linear OA(8¹⁴¹, 32769, F₈, 33) (dual of [32769, 32628, 34]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 32769Â | 8¹⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(129, 162, 4096)-Net in Base 8 — Constructive

(129, 162, 4096)-net in base 8, using

net defined by OOA [i] based on OOA(8¹⁶², 4096, S₈, 33, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(8¹⁶², 65537, S₈, 33), using
  - discarding parts of the base [i] based on linear OA(16¹²¹, 65537, F₁₆, 33) (dual of [65537, 65416, 34]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 65537Â | 16⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(129, 162, 68199)-Net over F₈ — Digital

Digital (129, 162, 68199)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 162, large)-Net in Base 8 — Upper bound on s

There is no (129, 162, large)-net in base 8, because

31 times m-reduction [i] would yield (129, 131, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]