Best Known (29, 47, s)-Nets in Base 8
(29, 47, 256)-Net over F8 — Constructive and digital
Digital (29, 47, 256)-net over F8, using
- 1 times m-reduction [i] based on digital (29, 48, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 24, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 24, 128)-net over F64, using
(29, 47, 300)-Net in Base 8 — Constructive
(29, 47, 300)-net in base 8, using
- 1 times m-reduction [i] based on (29, 48, 300)-net in base 8, using
- trace code for nets [i] based on (5, 24, 150)-net in base 64, using
- 4 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- base change [i] based on digital (1, 24, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 24, 150)-net over F128, using
- 4 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- trace code for nets [i] based on (5, 24, 150)-net in base 64, using
(29, 47, 375)-Net over F8 — Digital
Digital (29, 47, 375)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(847, 375, F8, 18) (dual of [375, 328, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(847, 518, F8, 18) (dual of [518, 471, 19]-code), using
- construction XX applied to C1 = C([57,73]), C2 = C([59,74]), C3 = C1 + C2 = C([59,73]), and C∩ = C1 ∩ C2 = C([57,74]) [i] based on
- linear OA(843, 511, F8, 17) (dual of [511, 468, 18]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {57,58,…,73}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(843, 511, F8, 16) (dual of [511, 468, 17]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {59,60,…,74}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(846, 511, F8, 18) (dual of [511, 465, 19]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {57,58,…,74}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(840, 511, F8, 15) (dual of [511, 471, 16]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {59,60,…,73}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(81, 4, F8, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- Reed–Solomon code RS(7,8) [i]
- discarding factors / shortening the dual code based on linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- 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 XX applied to C1 = C([57,73]), C2 = C([59,74]), C3 = C1 + C2 = C([59,73]), and C∩ = C1 ∩ C2 = C([57,74]) [i] based on
- discarding factors / shortening the dual code based on linear OA(847, 518, F8, 18) (dual of [518, 471, 19]-code), using
(29, 47, 30811)-Net in Base 8 — Upper bound on s
There is no (29, 47, 30812)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2 787845 827028 041710 954888 808432 055786 019599 > 847 [i]