Best Known (167−125, 167, s)-Nets in Base 4
(167−125, 167, 56)-Net over F4 — Constructive and digital
Digital (42, 167, 56)-net over F4, using
- t-expansion [i] based on digital (33, 167, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(167−125, 167, 75)-Net over F4 — Digital
Digital (42, 167, 75)-net over F4, using
- t-expansion [i] based on digital (40, 167, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(167−125, 167, 187)-Net over F4 — Upper bound on s (digital)
There is no digital (42, 167, 188)-net over F4, because
- 1 times m-reduction [i] would yield digital (42, 166, 188)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4166, 188, F4, 124) (dual of [188, 22, 125]-code), but
- construction Y1 [i] would yield
- linear OA(4165, 176, F4, 124) (dual of [176, 11, 125]-code), but
- residual code [i] would yield linear OA(441, 51, F4, 31) (dual of [51, 10, 32]-code), but
- “Gur†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(441, 51, F4, 31) (dual of [51, 10, 32]-code), but
- OA(422, 188, S4, 12), but
- discarding factors would yield OA(422, 164, S4, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 18 187369 733464 > 422 [i]
- discarding factors would yield OA(422, 164, S4, 12), but
- linear OA(4165, 176, F4, 124) (dual of [176, 11, 125]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4166, 188, F4, 124) (dual of [188, 22, 125]-code), but
(167−125, 167, 278)-Net in Base 4 — Upper bound on s
There is no (42, 167, 279)-net in base 4, because
- 1 times m-reduction [i] would yield (42, 166, 279)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 10313 105914 381279 196413 927581 878032 998449 899758 473432 670037 348366 390963 254028 060738 953023 776540 871200 > 4166 [i]