MinT - Best Known (29−4, 29, s)-Nets in Base 5

Best Known (29−4, 29, s)-Nets in Base 5

(29−4, 29, 976572)-Net over F₅ — Constructive and digital

Digital (25, 29, 976572)-net over F₅, using

net defined by OOA [i] based on linear OOA(5²⁹, 976572, F₅, 4, 4) (dual of [(976572, 4), 3906259, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(5²⁹, 976572, F₅, 3, 4) (dual of [(976572, 3), 2929687, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(5²⁹, 1953144, F₅, 4) (dual of [1953144, 1953115, 5]-code), using
    - constructionÂ X applied to C_e(3) âŠ‚ C_e(1) [i] based on
      1. linear OA(5²⁸, 1953125, F₅, 4) (dual of [1953125, 1953097, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1953124Â = 5⁹âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
      2. linear OA(5¹⁰, 1953125, F₅, 2) (dual of [1953125, 1953115, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 1953124Â = 5⁹âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
      3. linear OA(5¹, 19, F₅, 1) (dual of [19, 18, 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]

(29−4, 29, 2594383)-Net over F₅ — Digital

Digital (25, 29, 2594383)-net over F₅, using

net defined by OOA [i] based on linear OOA(5²⁹, 2594383, F₅, 4, 4) (dual of [(2594383, 4), 10377503, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(5²⁹, 2594383, F₅, 3, 4) (dual of [(2594383, 3), 7783120, 5]-NRT-code), using
  - Gilbertâ€“VarÅ¡amov bound [i]

(29−4, 29, large)-Net in Base 5 — Upper bound on s

There is no (25, 29, large)-net in base 5, because

2 times m-reduction [i] would yield (25, 27, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]