Best Known (74−39, 74, s)-Nets in Base 128
(74−39, 74, 504)-Net over F128 — Constructive and digital
Digital (35, 74, 504)-net over F128, using
- 3 times m-reduction [i] based on digital (35, 77, 504)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (5, 26, 216)-net over F128, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 5 and N(F) ≥ 216, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- digital (9, 51, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- digital (5, 26, 216)-net over F128, using
- (u, u+v)-construction [i] based on
(74−39, 74, 548)-Net in Base 128 — Constructive
(35, 74, 548)-net in base 128, using
- (u, u+v)-construction [i] based on
- (7, 26, 260)-net in base 128, using
- 6 times m-reduction [i] based on (7, 32, 260)-net in base 128, using
- base change [i] based on digital (3, 28, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- base change [i] based on digital (3, 28, 260)-net over F256, using
- 6 times m-reduction [i] based on (7, 32, 260)-net in base 128, using
- digital (9, 48, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- (7, 26, 260)-net in base 128, using
(74−39, 74, 1521)-Net over F128 — Digital
Digital (35, 74, 1521)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12874, 1521, F128, 39) (dual of [1521, 1447, 40]-code), using
- 1357 step Varšamov–Edel lengthening with (ri) = (12, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0, 1, 28 times 0, 1, 32 times 0, 1, 37 times 0, 1, 42 times 0, 1, 48 times 0, 1, 55 times 0, 1, 63 times 0, 1, 71 times 0, 1, 82 times 0, 1, 93 times 0, 1, 106 times 0, 1, 121 times 0, 1, 137 times 0, 1, 157 times 0, 1, 178 times 0) [i] based on linear OA(12839, 129, F128, 39) (dual of [129, 90, 40]-code or 129-arc in PG(38,128)), using
- extended Reed–Solomon code RSe(90,128) [i]
- the expurgated narrow-sense BCH-code C(I) with length 129 | 1282−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- 1357 step Varšamov–Edel lengthening with (ri) = (12, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0, 1, 28 times 0, 1, 32 times 0, 1, 37 times 0, 1, 42 times 0, 1, 48 times 0, 1, 55 times 0, 1, 63 times 0, 1, 71 times 0, 1, 82 times 0, 1, 93 times 0, 1, 106 times 0, 1, 121 times 0, 1, 137 times 0, 1, 157 times 0, 1, 178 times 0) [i] based on linear OA(12839, 129, F128, 39) (dual of [129, 90, 40]-code or 129-arc in PG(38,128)), using
(74−39, 74, 7789934)-Net in Base 128 — Upper bound on s
There is no (35, 74, 7789935)-net in base 128, because
- 1 times m-reduction [i] would yield (35, 73, 7789935)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 6703 916752 730554 554534 438340 726883 854351 161652 124608 801932 981697 506802 270941 603495 228713 087332 789754 056111 475484 587761 738366 910118 287161 655999 236751 462776 > 12873 [i]