MinT - Best Known (100−21, 100, s)-Nets in Base 27

Best Known (100−21, 100, s)-Nets in Base 27

(100−21, 100, 53290)-Net over F₂₇ — Constructive and digital

Digital (79, 100, 53290)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 146)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27¹⁹, 146, F₂₇, 10, 10) (dual of [(146, 10), 1441, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(27¹⁹, 730, F₂₇, 10) (dual of [730, 711, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(27¹⁹, 731, F₂₇, 10) (dual of [731, 712, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
        linear OA(27¹⁹, 729, F₂₇, 10) (dual of [729, 710, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(27¹⁷, 729, F₂₇, 9) (dual of [729, 712, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 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 (60, 81, 53144)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸¹, 53144, F₂₇, 21, 21) (dual of [(53144, 21), 1115943, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(27⁸¹, 531441, F₂₇, 21) (dual of [531441, 531360, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 27⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(100−21, 100, 53294)-Net in Base 27 — Constructive

(79, 100, 53294)-net in base 27, using

base change [i] based on digital (54, 75, 53294)-net over F₈₁, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 14, 150)-net over F₈₁, using
    - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (40, 61, 53144)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁶¹, 53144, F₈₁, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
      - OOA 10-folding and stacking with additional row [i] based on linear OA(81⁶¹, 531441, F₈₁, 21) (dual of [531441, 531380, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(100−21, 100, 4583029)-Net over F₂₇ — Digital

Digital (79, 100, 4583029)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(100−21, 100, large)-Net in Base 27 — Upper bound on s

There is no (79, 100, large)-net in base 27, because

19 times m-reduction [i] would yield (79, 81, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (100−21, 100, s)-Nets in Base 27

(100−21, 100, 53290)-Net over F27 — Constructive and digital

(100−21, 100, 53294)-Net in Base 27 — Constructive

(100−21, 100, 4583029)-Net over F27 — Digital

(100−21, 100, large)-Net in Base 27 — Upper bound on s

(100−21, 100, 53290)-Net over F₂₇ — Constructive and digital

(100−21, 100, 4583029)-Net over F₂₇ — Digital