MinT - Best Known (157−23, 157, s)-Nets in Base 8

Best Known (157−23, 157, s)-Nets in Base 8

(157−23, 157, 190678)-Net over F₈ — Constructive and digital

Digital (134, 157, 190678)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 16, 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 (118, 141, 190650)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴¹, 190650, F₈, 23, 23) (dual of [(190650, 23), 4384809, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(8¹⁴¹, 2097151, F₈, 23) (dual of [2097151, 2097010, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴¹, 2097152, F₈, 23) (dual of [2097152, 2097011, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(157−23, 157, 3601926)-Net over F₈ — Digital

Digital (134, 157, 3601926)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−23, 157, large)-Net in Base 8 — Upper bound on s

There is no (134, 157, large)-net in base 8, because

21 times m-reduction [i] would yield (134, 136, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]