MinT - Best Known (33−12, 33, s)-Nets in Base 256

Best Known (33−12, 33, s)-Nets in Base 256

(33−12, 33, 11438)-Net over F₂₅₆ — Constructive and digital

Digital (21, 33, 11438)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 515)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256¹⁰, 515, F₂₅₆, 6, 6) (dual of [(515, 6), 3080, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(256¹⁰, 515, F₂₅₆, 5, 6) (dual of [(515, 5), 2565, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(256³, 257, F₂₅₆, 5, 3) (dual of [(257, 5), 1282, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(5;1282,256) [i]
        linear OOA(256⁷, 258, F₂₅₆, 5, 6) (dual of [(258, 5), 1283, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (1, 7, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (11, 23, 10923)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256²³, 10923, F₂₅₆, 12, 12) (dual of [(10923, 12), 131053, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(256²³, 65538, F₂₅₆, 12) (dual of [65538, 65515, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(256²³, 65536, F₂₅₆, 12) (dual of [65536, 65513, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(256²¹, 65536, F₂₅₆, 11) (dual of [65536, 65515, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(33−12, 33, 322999)-Net over F₂₅₆ — Digital

Digital (21, 33, 322999)-net over F₂₅₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−12, 33, large)-Net in Base 256 — Upper bound on s

There is no (21, 33, large)-net in base 256, because

10 times m-reduction [i] would yield (21, 23, large)-net in base 256, but
- mutually orthogonal hypercube bound [i]

Best Known (33−12, 33, s)-Nets in Base 256

(33−12, 33, 11438)-Net over F256 — Constructive and digital

(33−12, 33, 322999)-Net over F256 — Digital

(33−12, 33, large)-Net in Base 256 — Upper bound on s

(33−12, 33, 11438)-Net over F₂₅₆ — Constructive and digital

(33−12, 33, 322999)-Net over F₂₅₆ — Digital