MinT - Best Known (37−4, 37, s)-Nets in Base 128

Best Known (37−4, 37, s)-Nets in Base 128

(37−4, 37, large)-Net over F₁₂₈ — Constructive and digital

Digital (33, 37, large)-net over F₁₂₈, using

t-expansion [i] based on digital (31, 37, large)-net over F₁₂₈, using
- 1 times m-reduction [i] based on digital (31, 38, large)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (4, 7, large)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128⁷, large, F₁₂₈, 3, 3), using
        appending kth column [i] based on linear OOA(128⁷, large, F₁₂₈, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁷, large, F₁₂₈, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 128²-ovoid cap in AG(3,Â 128) [i] based on linear OA(128⁴, 16385, F₁₂₈, 3) (dual of [16385, 16381, 4]-code or 16385-cap in PG(3,128)), using
        ovoid in PG(3,Â 128) [i]
    2. digital (24, 31, 4926250)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128³¹, 4926250, F₁₂₈, 9, 7) (dual of [(4926250, 9), 44336219, 8]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(128³¹, 4926251, F₁₂₈, 3, 7) (dual of [(4926251, 3), 14778722, 8]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(128⁶, 2130050, F₁₂₈, 3, 3) (dual of [(2130050, 3), 6390144, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(128⁶, 2130050, F₁₂₈, 2, 3) (dual of [(2130050, 2), 4260094, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁶, 2130050, F₁₂₈, 3) (dual of [2130050, 2130044, 4]-code or 2130050-cap in PG(5,128)), using
        large caps in PG(5,Â b) [i]
        linear OOA(128²⁵, 2796201, F₁₂₈, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(128²⁵, large, F₁₂₈, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 17895697Â | 128⁴âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(37−4, 37, large)-Net in Base 128 — Upper bound on s

There is no (33, 37, large)-net in base 128, because

2 times m-reduction [i] would yield (33, 35, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (37−4, 37, s)-Nets in Base 128

(37−4, 37, large)-Net over F128 — Constructive and digital

(37−4, 37, large)-Net in Base 128 — Upper bound on s

(37−4, 37, large)-Net over F₁₂₈ — Constructive and digital