MinT - Best Known (42, 54, s)-Nets in Base 16

Best Known (42, 54, s)-Nets in Base 16

(42, 54, 21879)-Net over F₁₆ — Constructive and digital

Digital (42, 54, 21879)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (34, 46, 21846)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁴⁶, 21846, F₁₆, 12, 12) (dual of [(21846, 12), 262106, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(16⁴⁶, 131076, F₁₆, 12) (dual of [131076, 131030, 13]-code), using
      - trace code [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]

(42, 54, 43692)-Net in Base 16 — Constructive

(42, 54, 43692)-net in base 16, using

base change [i] based on digital (24, 36, 43692)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64³⁶, 43692, F₆₄, 12, 12) (dual of [(43692, 12), 524268, 13]-NRT-code), using
  - OA 6-folding and stacking [i] based on linear OA(64³⁶, 262152, F₆₄, 12) (dual of [262152, 262116, 13]-code), using
    - discarding factors / shortening the dual code based on linear OA(64³⁶, 262155, F₆₄, 12) (dual of [262155, 262119, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(8) [i] based on
        linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(64²⁵, 262144, F₆₄, 9) (dual of [262144, 262119, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(64², 11, F₆₄, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(42, 54, 266727)-Net over F₁₆ — Digital

Digital (42, 54, 266727)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 54, large)-Net in Base 16 — Upper bound on s

There is no (42, 54, large)-net in base 16, because

10 times m-reduction [i] would yield (42, 44, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (42, 54, s)-Nets in Base 16

(42, 54, 21879)-Net over F16 — Constructive and digital

(42, 54, 43692)-Net in Base 16 — Constructive

(42, 54, 266727)-Net over F16 — Digital

(42, 54, large)-Net in Base 16 — Upper bound on s

(42, 54, 21879)-Net over F₁₆ — Constructive and digital

(42, 54, 266727)-Net over F₁₆ — Digital