Best Known (48, 48+19, s)-Nets in Base 8
(48, 48+19, 456)-Net over F8 — Constructive and digital
Digital (48, 67, 456)-net over F8, using
- 81 times duplication [i] based on digital (47, 66, 456)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 456, F8, 19, 19) (dual of [(456, 19), 8598, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(866, 4105, F8, 19) (dual of [4105, 4039, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(865, 4097, F8, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(866, 4105, F8, 19) (dual of [4105, 4039, 20]-code), using
- net defined by OOA [i] based on linear OOA(866, 456, F8, 19, 19) (dual of [(456, 19), 8598, 20]-NRT-code), using
(48, 48+19, 576)-Net in Base 8 — Constructive
(48, 67, 576)-net in base 8, using
- 1 times m-reduction [i] based on (48, 68, 576)-net in base 8, using
- trace code for nets [i] based on (14, 34, 288)-net in base 64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on (14, 35, 288)-net in base 64, using
- trace code for nets [i] based on (14, 34, 288)-net in base 64, using
(48, 48+19, 3278)-Net over F8 — Digital
Digital (48, 67, 3278)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(867, 3278, F8, 19) (dual of [3278, 3211, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 4107, F8, 19) (dual of [4107, 4040, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(865, 4097, F8, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 4107, F8, 19) (dual of [4107, 4040, 20]-code), using
(48, 48+19, 2484919)-Net in Base 8 — Upper bound on s
There is no (48, 67, 2484920)-net in base 8, because
- 1 times m-reduction [i] would yield (48, 66, 2484920)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 401734 699511 635562 609021 654214 272563 690881 116172 027961 421774 > 866 [i]