MinT - Best Known (37−12, 37, s)-Nets in Base 8

Best Known (37−12, 37, s)-Nets in Base 8

(37−12, 37, 260)-Net over F₈ — Constructive and digital

Digital (25, 37, 260)-net over F₈, using

1 times m-reduction [i] based on digital (25, 38, 260)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 12, 130)-net over F₈, using
    - trace code for nets [i] based on 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 (13, 26, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 13, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(37−12, 37, 516)-Net in Base 8 — Constructive

(25, 37, 516)-net in base 8, using

1 times m-reduction [i] based on (25, 38, 516)-net in base 8, using
- trace code for nets [i] based on (6, 19, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (6, 20, 258)-net in base 64, using
    - base change [i] based on digital (1, 15, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(37−12, 37, 775)-Net over F₈ — Digital

Digital (25, 37, 775)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³⁷, 775, F₈, 12) (dual of [775, 738, 13]-code), using
- 252 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0, 1, 75 times 0, 1, 117 times 0) [i] based on linear OA(8³¹, 517, F₈, 12) (dual of [517, 486, 13]-code), using
  - constructionÂ XX applied to C₁Â =Â C([510,9]), C₂Â =Â C([0,10]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,9]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([510,10]) [i] based on
    1. linear OA(8²⁸, 511, F₈, 11) (dual of [511, 483, 12]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    2. linear OA(8²⁸, 511, F₈, 11) (dual of [511, 483, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    3. linear OA(8³¹, 511, F₈, 12) (dual of [511, 480, 13]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,10}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    4. linear OA(8²⁵, 511, F₈, 10) (dual of [511, 486, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    5. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    6. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 1]-code) (see above)

(37−12, 37, 158551)-Net in Base 8 — Upper bound on s

There is no (25, 37, 158552)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2596 218644 889416 292803 969013 110275 > 8³⁷ [i]

Best Known (37−12, 37, s)-Nets in Base 8

(37−12, 37, 260)-Net over F8 — Constructive and digital

(37−12, 37, 516)-Net in Base 8 — Constructive

(37−12, 37, 775)-Net over F8 — Digital

(37−12, 37, 158551)-Net in Base 8 — Upper bound on s

(37−12, 37, 260)-Net over F₈ — Constructive and digital

(37−12, 37, 775)-Net over F₈ — Digital