MinT - Best Known (57, 79, s)-Nets in Base 8

Best Known (57, 79, s)-Nets in Base 8

(57, 79, 419)-Net over F₈ — Constructive and digital

Digital (57, 79, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(10,64) in PG(20,8)) for nets [i] based on digital (0, 11, 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 (36, 58, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 29, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(57, 79, 576)-Net in Base 8 — Constructive

(57, 79, 576)-net in base 8, using

5 times m-reduction [i] based on (57, 84, 576)-net in base 8, using
- trace code for nets [i] based on (15, 42, 288)-net in base 64, using
  - base change [i] based on digital (9, 36, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(57, 79, 3935)-Net over F₈ — Digital

Digital (57, 79, 3935)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁹, 3935, F₈, 22) (dual of [3935, 3856, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷⁹, 4107, F₈, 22) (dual of [4107, 4028, 23]-code), using
  - constructionÂ XX applied to C_e(21) âŠ‚ C_e(19) âŠ‚ C_e(18) [i] based on
    1. linear OA(8⁷⁷, 4096, F₈, 22) (dual of [4096, 4019, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(8⁶⁹, 4096, F₈, 20) (dual of [4096, 4027, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    3. linear OA(8⁶⁵, 4096, F₈, 19) (dual of [4096, 4031, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    4. linear OA(8¹, 10, F₈, 1) (dual of [10, 9, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, 511, F₈, 1) (dual of [511, 510, 2]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,0], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
    5. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
      - dual of repetition code with length 1 [i]

(57, 79, 2146559)-Net in Base 8 — Upper bound on s

There is no (57, 79, 2146560)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 220856 762589 332307 066403 276370 158795 782098 235657 747531 015253 853313 675041 > 8⁷⁹ [i]

Best Known (57, 79, s)-Nets in Base 8

(57, 79, 419)-Net over F8 — Constructive and digital

(57, 79, 576)-Net in Base 8 — Constructive

(57, 79, 3935)-Net over F8 — Digital

(57, 79, 2146559)-Net in Base 8 — Upper bound on s

(57, 79, 419)-Net over F₈ — Constructive and digital

(57, 79, 3935)-Net over F₈ — Digital