MinT - Best Known (71−28, 71, s)-Nets in Base 128

Best Known (71−28, 71, s)-Nets in Base 128

(71−28, 71, 1320)-Net over F₁₂₈ — Constructive and digital

Digital (43, 71, 1320)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (43, 72, 1320)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 15, 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 (28, 57, 1170)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁵⁷, 1170, F₁₂₈, 29, 29) (dual of [(1170, 29), 33873, 30]-NRT-code), using
      - OOA 14-folding and stacking with additional row [i] based on linear OA(128⁵⁷, 16381, F₁₂₈, 29) (dual of [16381, 16324, 30]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁵⁷, 16384, F₁₂₈, 29) (dual of [16384, 16327, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(71−28, 71, 4682)-Net in Base 128 — Constructive

(43, 71, 4682)-net in base 128, using

1 times m-reduction [i] based on (43, 72, 4682)-net in base 128, using
- base change [i] based on digital (34, 63, 4682)-net over F₂₅₆, using
  - 256² times duplication [i] based on digital (32, 61, 4682)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁶¹, 4682, F₂₅₆, 29, 29) (dual of [(4682, 29), 135717, 30]-NRT-code), using
      - OOA 14-folding and stacking with additional row [i] based on linear OA(256⁶¹, 65549, F₂₅₆, 29) (dual of [65549, 65488, 30]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁶¹, 65550, F₂₅₆, 29) (dual of [65550, 65489, 30]-code), using
        constructionÂ X applied to C_e(28) âŠ‚ C_e(23) [i] based on
        linear OA(256⁵⁷, 65536, F₂₅₆, 29) (dual of [65536, 65479, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(256⁴⁷, 65536, F₂₅₆, 24) (dual of [65536, 65489, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
        linear OA(256⁴, 14, F₂₅₆, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁴, 256, F₂₅₆, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
        Reedâ€“Solomon code RS(252,256) [i]

(71−28, 71, 29921)-Net over F₁₂₈ — Digital

Digital (43, 71, 29921)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−28, 71, large)-Net in Base 128 — Upper bound on s

There is no (43, 71, large)-net in base 128, because

26 times m-reduction [i] would yield (43, 45, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (71−28, 71, s)-Nets in Base 128

(71−28, 71, 1320)-Net over F128 — Constructive and digital

(71−28, 71, 4682)-Net in Base 128 — Constructive

(71−28, 71, 29921)-Net over F128 — Digital

(71−28, 71, large)-Net in Base 128 — Upper bound on s

(71−28, 71, 1320)-Net over F₁₂₈ — Constructive and digital

(71−28, 71, 29921)-Net over F₁₂₈ — Digital