MinT - Best Known (31, 58, s)-Nets in Base 256

Best Known (31, 58, s)-Nets in Base 256

(31, 58, 5042)-Net over F₂₅₆ — Constructive and digital

Digital (31, 58, 5042)-net over F₂₅₆, using

256² times duplication [i] based on digital (29, 56, 5042)-net over F₂₅₆, using
- net defined by OOA [i] based on linear OOA(256⁵⁶, 5042, F₂₅₆, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
  - OOA 13-folding and stacking with additional row [i] based on linear OA(256⁵⁶, 65547, F₂₅₆, 27) (dual of [65547, 65491, 28]-code), using
    - discarding factors / shortening the dual code based on linear OA(256⁵⁶, 65548, F₂₅₆, 27) (dual of [65548, 65492, 28]-code), using
      - constructionÂ X applied to C([0,13]) âŠ‚ C([0,11]) [i] based on
        linear OA(256⁵³, 65537, F₂₅₆, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(256⁴⁵, 65537, F₂₅₆, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(31, 58, 21210)-Net over F₂₅₆ — Digital

Digital (31, 58, 21210)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁸, 21210, F₂₅₆, 3, 27) (dual of [(21210, 3), 63572, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(256⁵⁸, 21851, F₂₅₆, 3, 27) (dual of [(21851, 3), 65495, 28]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(256⁵⁸, 65553, F₂₅₆, 27) (dual of [65553, 65495, 28]-code), using
    - discarding factors / shortening the dual code based on linear OA(256⁵⁸, 65554, F₂₅₆, 27) (dual of [65554, 65496, 28]-code), using
      - constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
        linear OA(256⁵³, 65537, F₂₅₆, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(256⁴¹, 65537, F₂₅₆, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(256⁵, 17, F₂₅₆, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁵, 256, F₂₅₆, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
        Reedâ€“Solomon code RS(251,256) [i]

(31, 58, large)-Net in Base 256 — Upper bound on s

There is no (31, 58, large)-net in base 256, because

25 times m-reduction [i] would yield (31, 33, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (31, 58, s)-Nets in Base 256

(31, 58, 5042)-Net over F256 — Constructive and digital

(31, 58, 21210)-Net over F256 — Digital

(31, 58, large)-Net in Base 256 — Upper bound on s

(31, 58, 5042)-Net over F₂₅₆ — Constructive and digital

(31, 58, 21210)-Net over F₂₅₆ — Digital