MinT - Best Known (116−23, 116, s)-Nets in Base 16

Best Known (116−23, 116, s)-Nets in Base 16

(116−23, 116, 95328)-Net over F₁₆ — Constructive and digital

Digital (93, 116, 95328)-net over F₁₆, using

16³ times duplication [i] based on digital (90, 113, 95328)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16¹¹³, 95328, F₁₆, 23, 23) (dual of [(95328, 23), 2192431, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(16¹¹³, 1048609, F₁₆, 23) (dual of [1048609, 1048496, 24]-code), using
    - discarding factors / shortening the dual code based on linear OA(16¹¹³, 1048613, F₁₆, 23) (dual of [1048613, 1048500, 24]-code), using
      - constructionÂ X applied to C_e(22) âŠ‚ C_e(16) [i] based on
        linear OA(16¹⁰⁶, 1048576, F₁₆, 23) (dual of [1048576, 1048470, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(16⁷⁶, 1048576, F₁₆, 17) (dual of [1048576, 1048500, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(16⁷, 37, F₁₆, 5) (dual of [37, 30, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
        a code by Danev and Olsson [i]

(116−23, 116, 1348222)-Net over F₁₆ — Digital

Digital (93, 116, 1348222)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(116−23, 116, large)-Net in Base 16 — Upper bound on s

There is no (93, 116, large)-net in base 16, because

21 times m-reduction [i] would yield (93, 95, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]