Best Known (97−76, 97, s)-Nets in Base 4
(97−76, 97, 34)-Net over F4 — Constructive and digital
Digital (21, 97, 34)-net over F4, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- T5 from the second 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) = 21 and N(F) ≥ 34, using
(97−76, 97, 44)-Net over F4 — Digital
Digital (21, 97, 44)-net over F4, using
- net from sequence [i] based on digital (21, 43)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 44, using
(97−76, 97, 91)-Net over F4 — Upper bound on s (digital)
There is no digital (21, 97, 92)-net over F4, because
- 12 times m-reduction [i] would yield digital (21, 85, 92)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(485, 92, F4, 64) (dual of [92, 7, 65]-code), but
- construction Y1 [i] would yield
- linear OA(484, 88, F4, 64) (dual of [88, 4, 65]-code), but
- OA(47, 92, S4, 4), but
- discarding factors would yield OA(47, 61, S4, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 16654 > 47 [i]
- discarding factors would yield OA(47, 61, S4, 4), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(485, 92, F4, 64) (dual of [92, 7, 65]-code), but
(97−76, 97, 92)-Net in Base 4 — Upper bound on s
There is no (21, 97, 93)-net in base 4, because
- 14 times m-reduction [i] would yield (21, 83, 93)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(483, 93, S4, 62), but
- the linear programming bound shows that M ≥ 412 815986 320748 811220 279730 438614 536337 297851 038207 836160 / 3 482479 > 483 [i]
- extracting embedded orthogonal array [i] would yield OA(483, 93, S4, 62), but