MinT - Best Known (77−14, 77, s)-Nets in Base 128

Best Known (77−14, 77, s)-Nets in Base 128

(77−14, 77, 1946578)-Net over F₁₂₈ — Constructive and digital

Digital (63, 77, 1946578)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (17, 24, 748207)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 5, 49156)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128⁵, 49156, F₁₂₈, 3, 3) (dual of [(49156, 3), 147463, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(128⁵, 49156, F₁₂₈, 2, 3) (dual of [(49156, 2), 98307, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁵, 49156, F₁₂₈, 3) (dual of [49156, 49151, 4]-code or 49156-cap in PG(4,128)), using
        large caps in PG(4,Â b) [i]
    2. digital (12, 19, 699051)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128¹⁹, 699051, F₁₂₈, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(128¹⁹, 2097154, F₁₂₈, 7) (dual of [2097154, 2097135, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹⁹, 2097155, F₁₂₈, 7) (dual of [2097155, 2097136, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(128¹⁹, 2097152, F₁₂₈, 7) (dual of [2097152, 2097133, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(128¹⁶, 2097152, F₁₂₈, 6) (dual of [2097152, 2097136, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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]

(77−14, 77, 2397256)-Net in Base 128 — Constructive

(63, 77, 2397256)-net in base 128, using

128¹ times duplication [i] based on (62, 76, 2397256)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (15, 22, 2796200)-net in base 128, using
    - net defined by OOA [i] based on OOA(128²², 2796200, S₁₂₈, 7, 7), using
      - OOA 3-folding and stacking with additional row [i] based on OA(128²², 8388601, S₁₂₈, 7), using
        discarding factors based on OA(128²², large, S₁₂₈, 7), using
        discarding parts of the base [i] based on linear OA(256¹⁹, large, F₂₅₆, 7) (dual of [large, large−19, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
  2. (40, 54, 1198628)-net in base 128, using
    - (u,Â u+v)-construction [i] based on
      1. (1, 8, 257)-net in base 128, using
        base change [i] based on digital (0, 7, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [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]

(77−14, 77, large)-Net over F₁₂₈ — Digital

Digital (63, 77, large)-net over F₁₂₈, using

t-expansion [i] based on digital (56, 77, large)-net over F₁₂₈, using
- Gilbertâ€“VarÅ¡amov bound [i]

(77−14, 77, large)-Net in Base 128 — Upper bound on s

There is no (63, 77, large)-net in base 128, because

12 times m-reduction [i] would yield (63, 65, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (77−14, 77, s)-Nets in Base 128

(77−14, 77, 1946578)-Net over F128 — Constructive and digital

(77−14, 77, 2397256)-Net in Base 128 — Constructive

(77−14, 77, large)-Net over F128 — Digital

(77−14, 77, large)-Net in Base 128 — Upper bound on s

(77−14, 77, 1946578)-Net over F₁₂₈ — Constructive and digital

(77−14, 77, large)-Net over F₁₂₈ — Digital