Best Known (32, 32+76, s)-Nets in Base 4
(32, 32+76, 34)-Net over F4 — Constructive and digital
Digital (32, 108, 34)-net over F4, using
- t-expansion [i] based on digital (21, 108, 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
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
(32, 32+76, 43)-Net in Base 4 — Constructive
(32, 108, 43)-net in base 4, using
- t-expansion [i] based on (30, 108, 43)-net in base 4, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
(32, 32+76, 60)-Net over F4 — Digital
Digital (32, 108, 60)-net over F4, using
- t-expansion [i] based on digital (31, 108, 60)-net over F4, using
- net from sequence [i] based on digital (31, 59)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 31 and N(F) ≥ 60, using
- net from sequence [i] based on digital (31, 59)-sequence over F4, using
(32, 32+76, 219)-Net over F4 — Upper bound on s (digital)
There is no digital (32, 108, 220)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4108, 220, F4, 76) (dual of [220, 112, 77]-code), but
- construction Y1 [i] would yield
- OA(4107, 141, S4, 76), but
- the linear programming bound shows that M ≥ 94 704083 303464 777353 653894 455864 731569 878383 755711 840806 960834 620243 478072 712366 905131 794432 / 3301 695418 882126 135009 765625 > 4107 [i]
- linear OA(4112, 220, F4, 79) (dual of [220, 108, 80]-code), but
- discarding factors / shortening the dual code would yield linear OA(4112, 216, F4, 79) (dual of [216, 104, 80]-code), but
- construction Y1 [i] would yield
- OA(4111, 143, S4, 79), but
- the linear programming bound shows that M ≥ 237 737030 554406 831140 331461 504711 505012 046426 036658 538250 578745 045288 416863 000860 792142 692352 / 26 111742 312416 213213 385625 > 4111 [i]
- OA(4104, 216, S4, 73), but
- discarding factors would yield OA(4104, 150, S4, 73), but
- the linear programming bound shows that M ≥ 32 028783 264027 932827 838658 827950 132017 443030 031565 779091 184319 519613 927802 631343 679332 066747 830675 439616 / 70932 376441 373736 310653 589982 582096 950125 > 4104 [i]
- discarding factors would yield OA(4104, 150, S4, 73), but
- OA(4111, 143, S4, 79), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(4112, 216, F4, 79) (dual of [216, 104, 80]-code), but
- OA(4107, 141, S4, 76), but
- construction Y1 [i] would yield
(32, 32+76, 227)-Net in Base 4 — Upper bound on s
There is no (32, 108, 228)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 109957 007007 180486 918019 955219 694291 749254 287776 694200 017602 091120 > 4108 [i]