MinT - Best Known (83−19, 83, s)-Nets in Base 5

Best Known (83−19, 83, s)-Nets in Base 5

(83−19, 83, 356)-Net over F₅ — Constructive and digital

Digital (64, 83, 356)-net over F₅, using

5¹ times duplication [i] based on digital (63, 82, 356)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 24, 104)-net over F₅, using
    - trace code for nets [i] based on digital (3, 12, 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 (39, 58, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 29, 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]

(83−19, 83, 3356)-Net over F₅ — Digital

Digital (64, 83, 3356)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸³, 3356, F₅, 19) (dual of [3356, 3273, 20]-code), using
- 219 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 12 times 0, 1, 27 times 0, 1, 57 times 0, 1, 112 times 0) [i] based on linear OA(5⁷⁶, 3130, F₅, 19) (dual of [3130, 3054, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
    1. linear OA(5⁷⁶, 3125, F₅, 19) (dual of [3125, 3049, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(5⁷¹, 3125, F₅, 18) (dual of [3125, 3054, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(83−19, 83, 2421495)-Net in Base 5 — Upper bound on s

There is no (64, 83, 2421496)-net in base 5, because

1 times m-reduction [i] would yield (64, 82, 2421496)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2067 953558 780515 317881 196481 023804 801070 055064 872894 916065 > 5⁸² [i]