MinT - Best Known (98−15, 98, s)-Nets in Base 4

Best Known (98−15, 98, s)-Nets in Base 4

(98−15, 98, 9373)-Net over F₄ — Constructive and digital

Digital (83, 98, 9373)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (74, 89, 9363)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁸⁹, 9363, F₄, 15, 15) (dual of [(9363, 15), 140356, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(4⁸⁹, 65542, F₄, 15) (dual of [65542, 65453, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁸⁹, 65544, F₄, 15) (dual of [65544, 65455, 16]-code), using
        constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
        linear OA(4⁸⁹, 65536, F₄, 15) (dual of [65536, 65447, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(4⁸¹, 65536, F₄, 14) (dual of [65536, 65455, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(4⁰, 8, F₄, 0) (dual of [8, 8, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(98−15, 98, 58682)-Net over F₄ — Digital

Digital (83, 98, 58682)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁸, 58682, F₄, 15) (dual of [58682, 58584, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁹⁸, 65553, F₄, 15) (dual of [65553, 65455, 16]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4⁹, 17, F₄, 7) (dual of [17, 8, 8]-code), using
      - dual code (with bound on d by constructionÂ Y1) [i] based on
        linear OA(4⁸, 17, F₄, 6) (dual of [17, 9, 7]-code), using
        the BCH-code C(I) with length 17Â | 4⁴âˆ’1, defining interval IÂ =Â {1,4,7,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        nonexistence of linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), because
        bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. linear OA(4⁸⁹, 65536, F₄, 15) (dual of [65536, 65447, 16]-code), using
      - an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(98−15, 98, large)-Net in Base 4 — Upper bound on s

There is no (83, 98, large)-net in base 4, because

13 times m-reduction [i] would yield (83, 85, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (98−15, 98, s)-Nets in Base 4

(98−15, 98, 9373)-Net over F4 — Constructive and digital

(98−15, 98, 58682)-Net over F4 — Digital

(98−15, 98, large)-Net in Base 4 — Upper bound on s

(98−15, 98, 9373)-Net over F₄ — Constructive and digital

(98−15, 98, 58682)-Net over F₄ — Digital