MinT - Best Known (70, 70+163, s)-Nets in Base 3

Best Known (70, 70+163, s)-Nets in Base 3

(70, 70+163, 48)-Net over F₃ — Constructive and digital

Digital (70, 233, 48)-net over F₃, using

t-expansion [i] based on digital (45, 233, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(70, 70+163, 82)-Net over F₃ — Digital

Digital (70, 233, 82)-net over F₃, using

t-expansion [i] based on digital (69, 233, 82)-net over F₃, using
- net from sequence [i] based on digital (69, 81)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
    - Drinfeld modules of rank 1 [i]

(70, 70+163, 219)-Net over F₃ — Upper bound on s (digital)

There is no digital (70, 233, 220)-net over F₃, because

19 times m-reduction [i] would yield digital (70, 214, 220)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²¹⁴, 220, F₃, 144) (dual of [220, 6, 145]-code), but
  - residual code [i] would yield linear OA(3⁷⁰, 75, F₃, 48) (dual of [75, 5, 49]-code), but
    - â€œHNâ€ bound on codes from Brouwerâ€™s database [i]

(70, 70+163, 223)-Net in Base 3 — Upper bound on s

There is no (70, 233, 224)-net in base 3, because

14 times m-reduction [i] would yield (70, 219, 224)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²¹⁹, 224, S₃, 149), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 8335 248417 898089 038639 422182 220625 700315 950641 493051 894370 647422 773355 762538 053940 268612 352977 320694 855609 / 25 > 3²¹⁹ [i]