MinT - Best Known (46, 46+9, s)-Nets in Base 8

Best Known (46, 46+9, s)-Nets in Base 8

(46, 46+9, 524301)-Net over F₈ — Constructive and digital

Digital (46, 55, 524301)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (41, 50, 524287)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁰, 524287, F₈, 9, 9) (dual of [(524287, 9), 4718533, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(8⁵⁰, 2097149, F₈, 9) (dual of [2097149, 2097099, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁵⁰, 2097152, F₈, 9) (dual of [2097152, 2097102, 10]-code), using
        an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(46, 46+9, 2097179)-Net over F₈ — Digital

Digital (46, 55, 2097179)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁵, 2097179, F₈, 9) (dual of [2097179, 2097124, 10]-code), using
- constructionÂ XX applied to C_e(8) âŠ‚ C_e(4) âŠ‚ C_e(3) [i] based on
  1. linear OA(8⁵⁰, 2097152, F₈, 9) (dual of [2097152, 2097102, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
  2. linear OA(8²⁹, 2097152, F₈, 5) (dual of [2097152, 2097123, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  3. linear OA(8²², 2097152, F₈, 4) (dual of [2097152, 2097130, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
  4. linear OA(8⁴, 26, F₈, 3) (dual of [26, 22, 4]-code or 26-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]
  5. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
    - dual of repetition code with length 1 [i]

(46, 46+9, large)-Net in Base 8 — Upper bound on s

There is no (46, 55, large)-net in base 8, because

7 times m-reduction [i] would yield (46, 48, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]