MinT - Best Known (78−14, 78, s)-Nets in Base 4

Best Known (78−14, 78, s)-Nets in Base 4

(78−14, 78, 2346)-Net over F₄ — Constructive and digital

Digital (64, 78, 2346)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (57, 71, 2341)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷¹, 2341, F₄, 14, 14) (dual of [(2341, 14), 32703, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(4⁷¹, 16387, F₄, 14) (dual of [16387, 16316, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷¹, 16391, F₄, 14) (dual of [16391, 16320, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(4⁷¹, 16384, F₄, 14) (dual of [16384, 16313, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(4⁶⁴, 16384, F₄, 13) (dual of [16384, 16320, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(4⁰, 7, F₄, 0) (dual of [7, 7, 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]

(78−14, 78, 12857)-Net over F₄ — Digital

Digital (64, 78, 12857)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁸, 12857, F₄, 14) (dual of [12857, 12779, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷⁸, 16392, F₄, 14) (dual of [16392, 16314, 15]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4⁷, 8, F₄, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
      - dual of repetition code with length 8 [i]
    2. linear OA(4⁷¹, 16384, F₄, 14) (dual of [16384, 16313, 15]-code), using
      - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(78−14, 78, 5760570)-Net in Base 4 — Upper bound on s

There is no (64, 78, 5760571)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 91343 951990 878670 290142 620920 478603 751821 125376 > 4⁷⁸ [i]

Best Known (78−14, 78, s)-Nets in Base 4

(78−14, 78, 2346)-Net over F4 — Constructive and digital

(78−14, 78, 12857)-Net over F4 — Digital

(78−14, 78, 5760570)-Net in Base 4 — Upper bound on s

(78−14, 78, 2346)-Net over F₄ — Constructive and digital

(78−14, 78, 12857)-Net over F₄ — Digital