MinT - Best Known (34−13, 34, s)-Nets in Base 128

Best Known (34−13, 34, s)-Nets in Base 128

(34−13, 34, 2988)-Net over F₁₂₈ — Constructive and digital

Digital (21, 34, 2988)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 258)-net over F₁₂₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 6, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
2. digital (12, 25, 2730)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128²⁵, 2730, F₁₂₈, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(128²⁵, 16381, F₁₂₈, 13) (dual of [16381, 16356, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(128²⁵, 16384, F₁₂₈, 13) (dual of [16384, 16359, 14]-code), using
        an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(34−13, 34, 10924)-Net in Base 128 — Constructive

(21, 34, 10924)-net in base 128, using

128² times duplication [i] based on (19, 32, 10924)-net in base 128, using
- base change [i] based on digital (15, 28, 10924)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256²⁸, 10924, F₂₅₆, 13, 13) (dual of [(10924, 13), 141984, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(256²⁸, 65545, F₂₅₆, 13) (dual of [65545, 65517, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(256²⁸, 65548, F₂₅₆, 13) (dual of [65548, 65520, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,4]) [i] based on
        linear OA(256²⁵, 65537, F₂₅₆, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(256¹⁷, 65537, F₂₅₆, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(34−13, 34, 38909)-Net over F₁₂₈ — Digital

Digital (21, 34, 38909)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(34−13, 34, large)-Net in Base 128 — Upper bound on s

There is no (21, 34, large)-net in base 128, because

11 times m-reduction [i] would yield (21, 23, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (34−13, 34, s)-Nets in Base 128

(34−13, 34, 2988)-Net over F128 — Constructive and digital

(34−13, 34, 10924)-Net in Base 128 — Constructive

(34−13, 34, 38909)-Net over F128 — Digital

(34−13, 34, large)-Net in Base 128 — Upper bound on s

(34−13, 34, 2988)-Net over F₁₂₈ — Constructive and digital

(34−13, 34, 38909)-Net over F₁₂₈ — Digital