MinT - Best Known (171−58, 171, s)-Nets in Base 8

Best Known (171−58, 171, s)-Nets in Base 8

(171−58, 171, 513)-Net over F₈ — Constructive and digital

Digital (113, 171, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(85,64) in PG(170,8)) for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(171−58, 171, 576)-Net in Base 8 — Constructive

(113, 171, 576)-net in base 8, using

t-expansion [i] based on (108, 171, 576)-net in base 8, using
- 1 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
  - trace code for nets [i] based on (22, 86, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 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]

(171−58, 171, 1644)-Net over F₈ — Digital

Digital (113, 171, 1644)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁷¹, 1644, F₈, 58) (dual of [1644, 1473, 59]-code), using
- 1472 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 8 times 0, 1, 10 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 34 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 40 times 0, 1, 41 times 0, 1, 43 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 50 times 0, 1, 52 times 0, 1, 54 times 0, 1, 56 times 0) [i] based on linear OA(8⁵⁸, 59, F₈, 58) (dual of [59, 1, 59]-code or 59-arc in PG(57,8)), using
  - dual of repetition code with length 59 [i]

(171−58, 171, 352466)-Net in Base 8 — Upper bound on s

There is no (113, 171, 352467)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26815 629933 176747 052210 659759 398942 051895 868950 067714 988242 766616 247635 531282 969335 205845 861976 912115 136267 384803 453570 632191 077352 297513 494047 529387 896928 > 8¹⁷¹ [i]

Best Known (171−58, 171, s)-Nets in Base 8

(171−58, 171, 513)-Net over F8 — Constructive and digital

(171−58, 171, 576)-Net in Base 8 — Constructive

(171−58, 171, 1644)-Net over F8 — Digital

(171−58, 171, 352466)-Net in Base 8 — Upper bound on s

(171−58, 171, 513)-Net over F₈ — Constructive and digital

(171−58, 171, 1644)-Net over F₈ — Digital