MinT - Best Known (238−32, 238, s)-Nets in Base 4

Best Known (238−32, 238, s)-Nets in Base 4

(238−32, 238, 16401)-Net over F₄ — Constructive and digital

Digital (206, 238, 16401)-net over F₄, using

t-expansion [i] based on digital (205, 238, 16401)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 21, 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 (184, 217, 16384)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4²¹⁷, 16384, F₄, 33, 33) (dual of [(16384, 33), 540455, 34]-NRT-code), using
      - OOA 16-folding and stacking with additional row [i] based on linear OA(4²¹⁷, 262145, F₄, 33) (dual of [262145, 261928, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(238−32, 238, 229032)-Net over F₄ — Digital

Digital (206, 238, 229032)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁸, 229032, F₄, 32) (dual of [229032, 228794, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4²³⁸, 262228, F₄, 32) (dual of [262228, 261990, 33]-code), using
  - 1 times truncation [i] based on linear OA(4²³⁹, 262229, F₄, 33) (dual of [262229, 261990, 34]-code), using
    - constructionÂ X applied to C_e(32) âŠ‚ C_e(22) [i] based on
      1. linear OA(4²¹⁷, 262144, F₄, 33) (dual of [262144, 261927, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
      2. linear OA(4¹⁵⁴, 262144, F₄, 23) (dual of [262144, 261990, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(4²², 85, F₄, 9) (dual of [85, 63, 10]-code), using
        the BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â {−20,−17,−14,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(238−32, 238, large)-Net in Base 4 — Upper bound on s

There is no (206, 238, large)-net in base 4, because

30 times m-reduction [i] would yield (206, 208, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]