MinT - Best Known (67−18, 67, s)-Nets in Base 128

Best Known (67−18, 67, s)-Nets in Base 128

(67−18, 67, 233317)-Net over F₁₂₈ — Constructive and digital

Digital (49, 67, 233317)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (6, 15, 300)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 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 (1, 10, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)
2. digital (34, 52, 233017)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵², 233017, F₁₂₈, 18, 18) (dual of [(233017, 18), 4194254, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(128⁵², 2097153, F₁₂₈, 18) (dual of [2097153, 2097101, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁵², 2097155, F₁₂₈, 18) (dual of [2097155, 2097103, 19]-code), using
        constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
        linear OA(128⁵², 2097152, F₁₂₈, 18) (dual of [2097152, 2097100, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(128⁴⁹, 2097152, F₁₂₈, 17) (dual of [2097152, 2097103, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(67−18, 67, 932067)-Net in Base 128 — Constructive

(49, 67, 932067)-net in base 128, using

128¹ times duplication [i] based on (48, 66, 932067)-net in base 128, using
- t-expansion [i] based on (47, 66, 932067)-net in base 128, using
  - net defined by OOA [i] based on OOA(128⁶⁶, 932067, S₁₂₈, 21, 19), using
    - OOA 3-folding and stacking with additional row [i] based on OOA(128⁶⁶, 2796202, S₁₂₈, 3, 19), using
      - 1 times NRT-code embedding in larger space [i] based on OOA(128⁶³, 2796201, S₁₂₈, 3, 19), using
        OOA 3-folding [i] based on OA(128⁶³, large, S₁₂₈, 19), using
        discarding parts of the base [i] based on linear OA(256⁵⁵, large, F₂₅₆, 19) (dual of [large, large−55, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(67−18, 67, large)-Net over F₁₂₈ — Digital

Digital (49, 67, large)-net over F₁₂₈, using

128¹ times duplication [i] based on digital (48, 66, large)-net over F₁₂₈, using
- Gilbertâ€“VarÅ¡amov bound [i]

(67−18, 67, large)-Net in Base 128 — Upper bound on s

There is no (49, 67, large)-net in base 128, because

16 times m-reduction [i] would yield (49, 51, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (67−18, 67, s)-Nets in Base 128

(67−18, 67, 233317)-Net over F128 — Constructive and digital

(67−18, 67, 932067)-Net in Base 128 — Constructive

(67−18, 67, large)-Net over F128 — Digital

(67−18, 67, large)-Net in Base 128 — Upper bound on s

(67−18, 67, 233317)-Net over F₁₂₈ — Constructive and digital

(67−18, 67, large)-Net over F₁₂₈ — Digital