MinT - Best Known (138, 138+21, s)-Nets in Base 8

Best Known (138, 138+21, s)-Nets in Base 8

(138, 138+21, 838885)-Net over F₈ — Constructive and digital

Digital (138, 159, 838885)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (124, 145, 838860)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁵, 838860, F₈, 21, 21) (dual of [(838860, 21), 17615915, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(8¹⁴⁵, 8388601, F₈, 21) (dual of [8388601, 8388456, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(138, 138+21, large)-Net over F₈ — Digital

Digital (138, 159, large)-net over F₈, using

2 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(138, 138+21, large)-Net in Base 8 — Upper bound on s

There is no (138, 159, large)-net in base 8, because

19 times m-reduction [i] would yield (138, 140, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]