Best Known (82, 82+32, s)-Nets in Base 8
(82, 82+32, 514)-Net over F8 — Constructive and digital
Digital (82, 114, 514)-net over F8, using
- t-expansion [i] based on digital (81, 114, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (18, 34, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 17, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 17, 80)-net over F64, using
- digital (47, 80, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- digital (18, 34, 160)-net over F8, using
- (u, u+v)-construction [i] based on
(82, 82+32, 593)-Net in Base 8 — Constructive
(82, 114, 593)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (2, 18, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- (64, 96, 576)-net in base 8, using
- trace code for nets [i] based on (16, 48, 288)-net in base 64, using
- 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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
- base change [i] based on digital (9, 42, 288)-net over F128, using
- 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- trace code for nets [i] based on (16, 48, 288)-net in base 64, using
- digital (2, 18, 17)-net over F8, using
(82, 82+32, 4106)-Net over F8 — Digital
Digital (82, 114, 4106)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8114, 4106, F8, 32) (dual of [4106, 3992, 33]-code), using
- 5 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0) [i] based on linear OA(8113, 4100, F8, 32) (dual of [4100, 3987, 33]-code), using
- 1 times truncation [i] based on linear OA(8114, 4101, F8, 33) (dual of [4101, 3987, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(8113, 4096, F8, 33) (dual of [4096, 3983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(8109, 4096, F8, 31) (dual of [4096, 3987, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(81, 5, F8, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(8114, 4101, F8, 33) (dual of [4101, 3987, 34]-code), using
- 5 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0) [i] based on linear OA(8113, 4100, F8, 32) (dual of [4100, 3987, 33]-code), using
(82, 82+32, 2642136)-Net in Base 8 — Upper bound on s
There is no (82, 114, 2642137)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 8 959011 799162 336037 184422 540350 454326 141997 463662 261227 117094 060765 280417 772998 417096 587448 391278 118715 > 8114 [i]