MinT - Best Known (91−25, 91, s)-Nets in Base 64

Best Known (91−25, 91, s)-Nets in Base 64

(91−25, 91, 21975)-Net over F₆₄ — Constructive and digital

Digital (66, 91, 21975)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 130)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 6, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 12, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (48, 73, 21845)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁷³, 21845, F₆₄, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(64⁷³, 262141, F₆₄, 25) (dual of [262141, 262068, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁷³, 262144, F₆₄, 25) (dual of [262144, 262071, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(91−25, 91, 174764)-Net in Base 64 — Constructive

(66, 91, 174764)-net in base 64, using

base change [i] based on digital (53, 78, 174764)-net over F₁₂₈, using
- 128¹ times duplication [i] based on digital (52, 77, 174764)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁷⁷, 174764, F₁₂₈, 25, 25) (dual of [(174764, 25), 4369023, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(128⁷⁷, 2097169, F₁₂₈, 25) (dual of [2097169, 2097092, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁷⁷, 2097171, F₁₂₈, 25) (dual of [2097171, 2097094, 26]-code), using
        constructionÂ X applied to C_e(24) âŠ‚ C_e(19) [i] based on
        linear OA(128⁷³, 2097152, F₁₂₈, 25) (dual of [2097152, 2097079, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(128⁵⁸, 2097152, F₁₂₈, 20) (dual of [2097152, 2097094, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(128⁴, 19, F₁₂₈, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁴, 128, F₁₂₈, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
        Reedâ€“Solomon code RS(124,128) [i]

(91−25, 91, 1097638)-Net over F₆₄ — Digital

Digital (66, 91, 1097638)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−25, 91, large)-Net in Base 64 — Upper bound on s

There is no (66, 91, large)-net in base 64, because

23 times m-reduction [i] would yield (66, 68, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (91−25, 91, s)-Nets in Base 64

(91−25, 91, 21975)-Net over F64 — Constructive and digital

(91−25, 91, 174764)-Net in Base 64 — Constructive

(91−25, 91, 1097638)-Net over F64 — Digital

(91−25, 91, large)-Net in Base 64 — Upper bound on s

(91−25, 91, 21975)-Net over F₆₄ — Constructive and digital

(91−25, 91, 1097638)-Net over F₆₄ — Digital