MinT - Best Known (79−21, 79, s)-Nets in Base 27

Best Known (79−21, 79, s)-Nets in Base 27

(79−21, 79, 2054)-Net over F₂₇ — Constructive and digital

Digital (58, 79, 2054)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 86)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
    2. digital (2, 12, 48)-net over F₂₇, using
      - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
2. digital (40, 61, 1968)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶¹, 1968, F₂₇, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(27⁶¹, 19681, F₂₇, 21) (dual of [19681, 19620, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁶¹, 19683, F₂₇, 21) (dual of [19683, 19622, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(79−21, 79, 2085)-Net in Base 27 — Constructive

(58, 79, 2085)-net in base 27, using

27¹ times duplication [i] based on (57, 78, 2085)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (6, 16, 116)-net in base 27, using
    - base change [i] based on digital (2, 12, 116)-net over F₈₁, using
      - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
  2. digital (41, 62, 1969)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁶², 1969, F₂₇, 21, 21) (dual of [(1969, 21), 41287, 22]-NRT-code), using
      - OOA 10-folding and stacking with additional row [i] based on linear OA(27⁶², 19691, F₂₇, 21) (dual of [19691, 19629, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(27⁶¹, 19684, F₂₇, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(27⁵⁵, 19684, F₂₇, 19) (dual of [19684, 19629, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 27⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(27¹, 7, F₂₇, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(79−21, 79, 143963)-Net over F₂₇ — Digital

Digital (58, 79, 143963)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(79−21, 79, large)-Net in Base 27 — Upper bound on s

There is no (58, 79, large)-net in base 27, because

19 times m-reduction [i] would yield (58, 60, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (79−21, 79, s)-Nets in Base 27

(79−21, 79, 2054)-Net over F27 — Constructive and digital

(79−21, 79, 2085)-Net in Base 27 — Constructive

(79−21, 79, 143963)-Net over F27 — Digital

(79−21, 79, large)-Net in Base 27 — Upper bound on s

(79−21, 79, 2054)-Net over F₂₇ — Constructive and digital

(79−21, 79, 143963)-Net over F₂₇ — Digital