Best Known (79, 79+27, s)-Nets in Base 8
(79, 79+27, 630)-Net over F8 — Constructive and digital
Digital (79, 106, 630)-net over F8, using
- net defined by OOA [i] based on linear OOA(8106, 630, F8, 27, 27) (dual of [(630, 27), 16904, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8106, 8191, F8, 27) (dual of [8191, 8085, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 8194, F8, 27) (dual of [8194, 8088, 28]-code), using
- trace code [i] based on linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- trace code [i] based on linear OA(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 8194, F8, 27) (dual of [8194, 8088, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8106, 8191, F8, 27) (dual of [8191, 8085, 28]-code), using
(79, 79+27, 814)-Net in Base 8 — Constructive
(79, 106, 814)-net in base 8, using
- (u, u+v)-construction [i] based on
- (21, 34, 300)-net in base 8, using
- trace code for nets [i] based on (4, 17, 150)-net in base 64, using
- 4 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- base change [i] based on digital (1, 18, 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, 18, 150)-net over F128, using
- 4 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- trace code for nets [i] based on (4, 17, 150)-net in base 64, using
- (45, 72, 514)-net in base 8, using
- base change [i] based on digital (27, 54, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 27, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 27, 257)-net over F256, using
- base change [i] based on digital (27, 54, 514)-net over F16, using
- (21, 34, 300)-net in base 8, using
(79, 79+27, 8196)-Net over F8 — Digital
Digital (79, 106, 8196)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8106, 8196, F8, 27) (dual of [8196, 8090, 28]-code), using
- trace code [i] based on linear OA(6453, 4098, F64, 27) (dual of [4098, 4045, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(6453, 4096, F64, 27) (dual of [4096, 4043, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- trace code [i] based on linear OA(6453, 4098, F64, 27) (dual of [4098, 4045, 28]-code), using
(79, 79+27, large)-Net in Base 8 — Upper bound on s
There is no (79, 106, large)-net in base 8, because
- 25 times m-reduction [i] would yield (79, 81, large)-net in base 8, but