MinT - Best Known (103, 103+34, s)-Nets in Base 5

Best Known (103, 103+34, s)-Nets in Base 5

(103, 103+34, 412)-Net over F₅ — Constructive and digital

Digital (103, 137, 412)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (84, 118, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 59, 200)-net over F₂₅, using
    - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
        a function field by GarcÃa and Quoos [i]

(103, 103+34, 2966)-Net over F₅ — Digital

Digital (103, 137, 2966)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³⁷, 2966, F₅, 34) (dual of [2966, 2829, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹³⁷, 3136, F₅, 34) (dual of [3136, 2999, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(31) [i] based on
    1. linear OA(5¹³⁶, 3125, F₅, 34) (dual of [3125, 2989, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(5¹²⁶, 3125, F₅, 32) (dual of [3125, 2999, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(103, 103+34, 770467)-Net in Base 5 — Upper bound on s

There is no (103, 137, 770468)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 573980 343480 689032 950403 015919 376171 025141 487073 793180 449846 305006 529365 799375 232465 000181 151889 > 5¹³⁷ [i]