Best Known (103, 103+11, s)-Nets in Base 8
(103, 103+11, 4404092)-Net over F8 — Constructive and digital
Digital (103, 114, 4404092)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (27, 32, 1048652)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 3, 73)-net over F8, using
- digital (24, 29, 1048579)-net over F8, using
- net defined by OOA [i] based on linear OOA(829, 1048579, F8, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(829, 2097159, F8, 5) (dual of [2097159, 2097130, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(829, 2097152, F8, 5) (dual of [2097152, 2097123, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(822, 2097152, F8, 4) (dual of [2097152, 2097130, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(829, 2097159, F8, 5) (dual of [2097159, 2097130, 6]-code), using
- net defined by OOA [i] based on linear OOA(829, 1048579, F8, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- digital (71, 82, 3355440)-net over F8, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- digital (27, 32, 1048652)-net over F8, using
(103, 103+11, 5452594)-Net in Base 8 — Constructive
(103, 114, 5452594)-net in base 8, using
- (u, u+v)-construction [i] based on
- (27, 32, 2097154)-net in base 8, using
- net defined by OOA [i] based on OOA(832, 2097154, S8, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(832, 4194309, S8, 5), using
- discarding factors based on OA(832, 4194310, S8, 5), using
- trace code [i] based on OA(6416, 2097155, S64, 5), using
- discarding parts of the base [i] based on linear OA(12813, 2097155, F128, 5) (dual of [2097155, 2097142, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(12813, 2097155, F128, 5) (dual of [2097155, 2097142, 6]-code), using
- trace code [i] based on OA(6416, 2097155, S64, 5), using
- discarding factors based on OA(832, 4194310, S8, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(832, 4194309, S8, 5), using
- net defined by OOA [i] based on OOA(832, 2097154, S8, 5, 5), using
- digital (71, 82, 3355440)-net over F8, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- (27, 32, 2097154)-net in base 8, using
(103, 103+11, large)-Net over F8 — Digital
Digital (103, 114, large)-net over F8, using
- t-expansion [i] based on digital (100, 114, large)-net over F8, using
- 3 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 3 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
(103, 103+11, large)-Net in Base 8 — Upper bound on s
There is no (103, 114, large)-net in base 8, because
- 9 times m-reduction [i] would yield (103, 105, large)-net in base 8, but