MinT - Best Known (99, 123, s)-Nets in Base 8

Best Known (99, 123, s)-Nets in Base 8

(99, 123, 2759)-Net over F₈ — Constructive and digital

Digital (99, 123, 2759)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 17, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (82, 106, 2731)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁰⁶, 2731, F₈, 24, 24) (dual of [(2731, 24), 65438, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(8¹⁰⁶, 32772, F₈, 24) (dual of [32772, 32666, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁰⁶, 32773, F₈, 24) (dual of [32773, 32667, 25]-code), using
        1 times truncation [i] based on linear OA(8¹⁰⁷, 32774, F₈, 25) (dual of [32774, 32667, 26]-code), using
        constructionÂ X applied to C_e(24) âŠ‚ C_e(22) [i] based on
        linear OA(8¹⁰⁶, 32768, F₈, 25) (dual of [32768, 32662, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(8¹⁰¹, 32768, F₈, 23) (dual of [32768, 32667, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(8¹, 6, F₈, 1) (dual of [6, 5, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(99, 123, 5462)-Net in Base 8 — Constructive

(99, 123, 5462)-net in base 8, using

8³ times duplication [i] based on (96, 120, 5462)-net in base 8, using
- base change [i] based on digital (66, 90, 5462)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁰, 5462, F₁₆, 24, 24) (dual of [(5462, 24), 130998, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(16⁹⁰, 65544, F₁₆, 24) (dual of [65544, 65454, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁹⁰, 65545, F₁₆, 24) (dual of [65545, 65455, 25]-code), using
        constructionÂ X applied to C_e(23) âŠ‚ C_e(21) [i] based on
        linear OA(16⁸⁹, 65536, F₁₆, 24) (dual of [65536, 65447, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
        linear OA(16⁸¹, 65536, F₁₆, 22) (dual of [65536, 65455, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(16¹, 9, F₁₆, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(99, 123, 90987)-Net over F₈ — Digital

Digital (99, 123, 90987)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 123, large)-Net in Base 8 — Upper bound on s

There is no (99, 123, large)-net in base 8, because

22 times m-reduction [i] would yield (99, 101, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (99, 123, s)-Nets in Base 8

(99, 123, 2759)-Net over F8 — Constructive and digital

(99, 123, 5462)-Net in Base 8 — Constructive

(99, 123, 90987)-Net over F8 — Digital

(99, 123, large)-Net in Base 8 — Upper bound on s

(99, 123, 2759)-Net over F₈ — Constructive and digital

(99, 123, 90987)-Net over F₈ — Digital