MinT - Best Known (123−30, 123, s)-Nets in Base 5

Best Known (123−30, 123, s)-Nets in Base 5

(123−30, 123, 408)-Net over F₅ — Constructive and digital

Digital (93, 123, 408)-net over F₅, using

3 times m-reduction [i] based on digital (93, 126, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 63, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

(123−30, 123, 3117)-Net over F₅ — Digital

Digital (93, 123, 3117)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²³, 3117, F₅, 30) (dual of [3117, 2994, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹²³, 3132, F₅, 30) (dual of [3132, 3009, 31]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(5¹²¹, 3126, F₅, 31) (dual of [3126, 3005, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(5¹¹¹, 3126, F₅, 27) (dual of [3126, 3015, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    3. linear OA(5², 6, F₅, 2) (dual of [6, 4, 3]-code or 6-arc in PG(1,5)), using
      - extended Reedâ€“Solomon code RS_e(4,5) [i]
      - Hamming code H(2,5) [i]

(123−30, 123, 865476)-Net in Base 5 — Upper bound on s

There is no (93, 123, 865477)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 94 040328 612949 464119 936929 808403 062233 465821 408266 630959 621056 598114 344632 244055 211485 > 5¹²³ [i]