MinT - Best Known (89, 89+11, s)-Nets in Base 5

Best Known (89, 89+11, s)-Nets in Base 5

(89, 89+11, 1678921)-Net over F₅ — Constructive and digital

Digital (89, 100, 1678921)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (14, 19, 1201)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹⁹, 1201, F₅, 6, 5) (dual of [(1201, 6), 7187, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(5¹⁹, 1202, F₅, 2, 5) (dual of [(1202, 2), 2385, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(5⁵, 781, F₅, 2, 2) (dual of [(781, 2), 1557, 3]-NRT-code), using
        appending kth column [i] based on linear OA(5⁵, 781, F₅, 2) (dual of [781, 776, 3]-code), using
        Hamming code H(5,5) [i]
        linear OOA(5¹⁴, 601, F₅, 2, 5) (dual of [(601, 2), 1188, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
        trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (70, 81, 1677720)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸¹, 1677720, F₅, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁸¹, 8388601, F₅, 11) (dual of [8388601, 8388520, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁸¹, large, F₅, 11) (dual of [large, large−81, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(89, 89+11, large)-Net over F₅ — Digital

Digital (89, 100, large)-net over F₅, using

t-expansion [i] based on digital (88, 100, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁰, large, F₅, 12) (dual of [large, large−100, 13]-code), using
  - 9 times code embedding in larger space [i] based on linear OA(5⁹¹, large, F₅, 12) (dual of [large, large−91, 13]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(89, 89+11, large)-Net in Base 5 — Upper bound on s

There is no (89, 100, large)-net in base 5, because

9 times m-reduction [i] would yield (89, 91, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (89, 89+11, s)-Nets in Base 5

(89, 89+11, 1678921)-Net over F5 — Constructive and digital

(89, 89+11, large)-Net over F5 — Digital

(89, 89+11, large)-Net in Base 5 — Upper bound on s

(89, 89+11, 1678921)-Net over F₅ — Constructive and digital

(89, 89+11, large)-Net over F₅ — Digital