MinT - Best Known (42−17, 42, s)-Nets in Base 64

Best Known (42−17, 42, s)-Nets in Base 64

(42−17, 42, 592)-Net over F₆₄ — Constructive and digital

Digital (25, 42, 592)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (16, 33, 512)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³³, 512, F₆₄, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(64³³, 4097, F₆₄, 17) (dual of [4097, 4064, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(42−17, 42, 2049)-Net in Base 64 — Constructive

(25, 42, 2049)-net in base 64, using

base change [i] based on digital (19, 36, 2049)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128³⁶, 2049, F₁₂₈, 17, 17) (dual of [(2049, 17), 34797, 18]-NRT-code), using
  - OOA 8-folding and stacking with additional row [i] based on linear OA(128³⁶, 16393, F₁₂₈, 17) (dual of [16393, 16357, 18]-code), using
    - discarding factors / shortening the dual code based on linear OA(128³⁶, 16396, F₁₂₈, 17) (dual of [16396, 16360, 18]-code), using
      - constructionÂ X applied to C([0,8]) âŠ‚ C([0,6]) [i] based on
        linear OA(128³³, 16385, F₁₂₈, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(128²⁵, 16385, F₁₂₈, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(128³, 11, F₁₂₈, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
        discarding factors / shortening the dual code based on linear OA(128³, 128, F₁₂₈, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
        Reedâ€“Solomon code RS(125,128) [i]

(42−17, 42, 6031)-Net over F₆₄ — Digital

Digital (25, 42, 6031)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴², 6031, F₆₄, 17) (dual of [6031, 5989, 18]-code), using
- 1924 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 4 times 0, 1, 18 times 0, 1, 69 times 0, 1, 208 times 0, 1, 531 times 0, 1, 1088 times 0) [i] based on linear OA(64³³, 4098, F₆₄, 17) (dual of [4098, 4065, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
    1. linear OA(64³³, 4096, F₆₄, 17) (dual of [4096, 4063, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(64³¹, 4096, F₆₄, 16) (dual of [4096, 4065, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(42−17, 42, 6234)-Net in Base 64

(25, 42, 6234)-net in base 64, using

base change [i] based on digital (19, 36, 6234)-net over F₁₂₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128³⁶, 6234, F₁₂₈, 2, 17) (dual of [(6234, 2), 12432, 18]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(128³⁶, 8198, F₁₂₈, 2, 17) (dual of [(8198, 2), 16360, 18]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(128³⁶, 16396, F₁₂₈, 17) (dual of [16396, 16360, 18]-code), using
      - constructionÂ X applied to C([0,8]) âŠ‚ C([0,6]) [i] based on
        linear OA(128³³, 16385, F₁₂₈, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(128²⁵, 16385, F₁₂₈, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(128³, 11, F₁₂₈, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
        discarding factors / shortening the dual code based on linear OA(128³, 128, F₁₂₈, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
        Reedâ€“Solomon code RS(125,128) [i]

(42−17, 42, large)-Net in Base 64 — Upper bound on s

There is no (25, 42, large)-net in base 64, because

15 times m-reduction [i] would yield (25, 27, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]