MinT - Best Known (61, 61+197, s)-Nets in Base 4

Best Known (61, 61+197, s)-Nets in Base 4

(61, 61+197, 66)-Net over F₄ — Constructive and digital

Digital (61, 258, 66)-net over F₄, using

t-expansion [i] based on digital (49, 258, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(61, 61+197, 99)-Net over F₄ — Digital

Digital (61, 258, 99)-net over F₄, using

net from sequence [i] based on digital (61, 98)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
  - Drinfeld modules of rank 1 [i]

(61, 61+197, 254)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 258, 255)-net over F₄, because

13 times m-reduction [i] would yield digital (61, 245, 255)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁵, 255, F₄, 184) (dual of [255, 10, 185]-code), but
  - residual code [i] would yield linear OA(4⁶¹, 70, F₄, 46) (dual of [70, 9, 47]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(61, 61+197, 259)-Net in Base 4 — Upper bound on s

There is no (61, 258, 260)-net in base 4, because

2 times m-reduction [i] would yield (61, 256, 260)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²⁵⁶, 260, S₄, 195), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 858099 707516 326214 372737 599885 174152 158679 412517 913176 174307 932398 192897 924707 006515 319955 082681 819372 162038 923935 107254 640248 499964 580476 571753 536389 382144 / 49 > 4²⁵⁶ [i]