MinT - Best Known (81, 81+23, s)-Nets in Base 5

Best Known (81, 81+23, s)-Nets in Base 5

(81, 81+23, 424)-Net over F₅ — Constructive and digital

Digital (81, 104, 424)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (27, 38, 172)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (5, 10, 68)-net over F₅, using
      - linear code embedding found in a computer search [i]
    2. digital (17, 28, 104)-net over F₅, using
      - trace code for nets [i] based on digital (3, 14, 52)-net over F₂₅, using
        net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
        a function field by GarcÃa and Quoos [i]
2. digital (43, 66, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 33, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(81, 81+23, 4572)-Net over F₅ — Digital

Digital (81, 104, 4572)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁴, 4572, F₅, 23) (dual of [4572, 4468, 24]-code), using
- 1429 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 1, 0, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 48 times 0, 1, 93 times 0, 1, 155 times 0, 1, 215 times 0, 1, 260 times 0, 1, 292 times 0, 1, 318 times 0) [i] based on linear OA(5⁹¹, 3130, F₅, 23) (dual of [3130, 3039, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. linear OA(5⁹¹, 3125, F₅, 23) (dual of [3125, 3034, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(5⁸⁶, 3125, F₅, 22) (dual of [3125, 3039, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(81, 81+23, 4303822)-Net in Base 5 — Upper bound on s

There is no (81, 104, 4303823)-net in base 5, because

1 times m-reduction [i] would yield (81, 103, 4303823)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 986077 870515 594466 857910 820017 926322 410239 170836 145467 805050 011817 608213 > 5¹⁰³ [i]