MinT - Best Known (61−16, 61, s)-Nets in Base 8

Best Known (61−16, 61, s)-Nets in Base 8

(61−16, 61, 514)-Net over F₈ — Constructive and digital

Digital (45, 61, 514)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁶¹, 514, F₈, 16, 16) (dual of [(514, 16), 8163, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8⁶¹, 4112, F₈, 16) (dual of [4112, 4051, 17]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁶¹, 4116, F₈, 16) (dual of [4116, 4055, 17]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(11) [i] based on
      1. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. linear OA(8⁴¹, 4096, F₈, 12) (dual of [4096, 4055, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      3. linear OA(8⁴, 20, F₈, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,8)), using
        discarding factors / shortening the dual code based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]

(61−16, 61, 644)-Net in Base 8 — Constructive

(45, 61, 644)-net in base 8, using

1 times m-reduction [i] based on (45, 62, 644)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 16, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 8, 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. (29, 46, 514)-net in base 8, using
    - trace code for nets [i] based on (6, 23, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
        base change [i] based on digital (0, 18, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(61−16, 61, 4539)-Net over F₈ — Digital

Digital (45, 61, 4539)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶¹, 4539, F₈, 16) (dual of [4539, 4478, 17]-code), using
- 438 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 35 times 0, 1, 117 times 0, 1, 276 times 0) [i] based on linear OA(8⁵⁶, 4096, F₈, 16) (dual of [4096, 4040, 17]-code), using
  - 1 times truncation [i] based on linear OA(8⁵⁷, 4097, F₈, 17) (dual of [4097, 4040, 18]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(61−16, 61, 4136687)-Net in Base 8 — Upper bound on s

There is no (45, 61, 4136688)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12 259981 872817 274933 448578 521297 806584 103375 540267 278287 > 8⁶¹ [i]