MinT - Best Known (61−14, 61, s)-Nets in Base 128

Best Known (61−14, 61, s)-Nets in Base 128

(61−14, 61, 1198521)-Net over F₁₂₈ — Constructive and digital

Digital (47, 61, 1198521)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (39, 53, 1198371)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵³, 1198371, F₁₂₈, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(128⁵³, 8388597, F₁₂₈, 14) (dual of [8388597, 8388544, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁵³, large, F₁₂₈, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9256395Â | 128⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(61−14, 61, 1220216)-Net in Base 128 — Constructive

(47, 61, 1220216)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (8, 15, 21845)-net in base 128, using
  - net defined by OOA [i] based on OOA(128¹⁵, 21845, S₁₂₈, 7, 7), using
    - OOA 3-folding and stacking with additional row [i] based on OA(128¹⁵, 65536, S₁₂₈, 7), using
      - discarding factors based on OA(128¹⁵, 65538, S₁₂₈, 7), using
        discarding parts of the base [i] based on linear OA(256¹³, 65538, F₂₅₆, 7) (dual of [65538, 65525, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(256¹¹, 65536, F₂₅₆, 6) (dual of [65536, 65525, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. (32, 46, 1198371)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁴⁶, 1198371, S₁₂₈, 14, 14), using
    - OA 7-folding and stacking [i] based on OA(128⁴⁶, 8388597, S₁₂₈, 14), using
      - discarding factors based on OA(128⁴⁶, large, S₁₂₈, 14), using
        discarding parts of the base [i] based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(61−14, 61, large)-Net over F₁₂₈ — Digital

Digital (47, 61, large)-net over F₁₂₈, using

t-expansion [i] based on digital (46, 61, large)-net over F₁₂₈, using
- 2 times m-reduction [i] based on digital (46, 63, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(61−14, 61, large)-Net in Base 128 — Upper bound on s

There is no (47, 61, large)-net in base 128, because

12 times m-reduction [i] would yield (47, 49, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (61−14, 61, s)-Nets in Base 128

(61−14, 61, 1198521)-Net over F128 — Constructive and digital

(61−14, 61, 1220216)-Net in Base 128 — Constructive

(61−14, 61, large)-Net over F128 — Digital

(61−14, 61, large)-Net in Base 128 — Upper bound on s

(61−14, 61, 1198521)-Net over F₁₂₈ — Constructive and digital

(61−14, 61, large)-Net over F₁₂₈ — Digital