Best Known (55, 89, s)-Nets in Base 8
(55, 89, 354)-Net over F8 — Constructive and digital
Digital (55, 89, 354)-net over F8, using
- 7 times m-reduction [i] based on digital (55, 96, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 177)-net over F64, using
(55, 89, 384)-Net in Base 8 — Constructive
(55, 89, 384)-net in base 8, using
- 1 times m-reduction [i] based on (55, 90, 384)-net in base 8, using
- trace code for nets [i] based on (10, 45, 192)-net in base 64, using
- 4 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- base change [i] based on digital (3, 42, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 42, 192)-net over F128, using
- 4 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- trace code for nets [i] based on (10, 45, 192)-net in base 64, using
(55, 89, 533)-Net over F8 — Digital
Digital (55, 89, 533)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(889, 533, F8, 34) (dual of [533, 444, 35]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0) [i] based on linear OA(888, 515, F8, 34) (dual of [515, 427, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(888, 512, F8, 34) (dual of [512, 424, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(885, 512, F8, 33) (dual of [512, 427, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 17 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0) [i] based on linear OA(888, 515, F8, 34) (dual of [515, 427, 35]-code), using
(55, 89, 54790)-Net in Base 8 — Upper bound on s
There is no (55, 89, 54791)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 237 170155 867027 134653 518307 698932 901422 473586 698757 584391 968348 607640 696670 056000 > 889 [i]