Best Known (51, 51+20, s)-Nets in Base 8
(51, 51+20, 410)-Net over F8 — Constructive and digital
Digital (51, 71, 410)-net over F8, using
- 82 times duplication [i] based on digital (49, 69, 410)-net over F8, using
- net defined by OOA [i] based on linear OOA(869, 410, F8, 20, 20) (dual of [(410, 20), 8131, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(869, 4100, F8, 20) (dual of [4100, 4031, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(869, 4096, F8, 20) (dual of [4096, 4027, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(865, 4096, F8, 19) (dual of [4096, 4031, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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(19) ⊂ Ce(18) [i] based on
- OA 10-folding and stacking [i] based on linear OA(869, 4100, F8, 20) (dual of [4100, 4031, 21]-code), using
- net defined by OOA [i] based on linear OOA(869, 410, F8, 20, 20) (dual of [(410, 20), 8131, 21]-NRT-code), using
(51, 51+20, 576)-Net in Base 8 — Constructive
(51, 71, 576)-net in base 8, using
- 81 times duplication [i] based on (50, 70, 576)-net in base 8, using
- t-expansion [i] based on (49, 70, 576)-net in base 8, using
- trace code for nets [i] based on (14, 35, 288)-net in base 64, using
- base change [i] based on digital (9, 30, 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, 30, 288)-net over F128, using
- trace code for nets [i] based on (14, 35, 288)-net in base 64, using
- t-expansion [i] based on (49, 70, 576)-net in base 8, using
(51, 51+20, 3497)-Net over F8 — Digital
Digital (51, 71, 3497)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(871, 3497, F8, 20) (dual of [3497, 3426, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(871, 4107, F8, 20) (dual of [4107, 4036, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(869, 4096, F8, 20) (dual of [4096, 4027, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(857, 4096, F8, 17) (dual of [4096, 4039, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(81, 10, F8, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(871, 4107, F8, 20) (dual of [4107, 4036, 21]-code), using
(51, 51+20, 1670381)-Net in Base 8 — Upper bound on s
There is no (51, 71, 1670382)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 13164 088560 843804 897690 780216 213217 033834 075452 628342 706737 294486 > 871 [i]