MinT - Best Known (234−25, 234, s)-Nets in Base 4

Best Known (234−25, 234, s)-Nets in Base 4

(234−25, 234, 699067)-Net over F₄ — Constructive and digital

Digital (209, 234, 699067)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 17, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (192, 217, 699050)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²¹⁷, 699050, F₄, 25, 25) (dual of [(699050, 25), 17476033, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(4²¹⁷, 8388601, F₄, 25) (dual of [8388601, 8388384, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²¹⁷, large, F₄, 25) (dual of [large, large−217, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(234−25, 234, 4194318)-Net over F₄ — Digital

Digital (209, 234, 4194318)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²³⁴, 4194318, F₄, 2, 25) (dual of [(4194318, 2), 8388402, 26]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(4¹⁷, 17, F₄, 2, 12) (dual of [(17, 2), 17, 13]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,21P) [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(4²¹⁷, 4194301, F₄, 2, 25) (dual of [(4194301, 2), 8388385, 26]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4²¹⁷, 8388602, F₄, 25) (dual of [8388602, 8388385, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²¹⁷, large, F₄, 25) (dual of [large, large−217, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(234−25, 234, large)-Net in Base 4 — Upper bound on s

There is no (209, 234, large)-net in base 4, because

23 times m-reduction [i] would yield (209, 211, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]