MinT - Best Known (239−178, 239, s)-Nets in Base 4

Best Known (239−178, 239, s)-Nets in Base 4

(239−178, 239, 66)-Net over F₄ — Constructive and digital

Digital (61, 239, 66)-net over F₄, using

t-expansion [i] based on digital (49, 239, 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]

(239−178, 239, 99)-Net over F₄ — Digital

Digital (61, 239, 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]

(239−178, 239, 267)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 239, 268)-net over F₄, because

2 times m-reduction [i] would yield digital (61, 237, 268)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁷, 268, F₄, 176) (dual of [268, 31, 177]-code), but
  - residual code [i] would yield OA(4⁶¹, 91, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 115457 192596 305257 505080 897617 251271 932542 088555 636975 468544 / 21352 537403 048391 328525 > 4⁶¹ [i]

(239−178, 239, 398)-Net in Base 4 — Upper bound on s

There is no (61, 239, 399)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 883031 636280 014254 329413 013917 421198 548688 152726 013957 057246 605500 894614 944613 258783 135097 800070 050451 582152 891978 613894 251475 718073 998547 900496 > 4²³⁹ [i]