Best Known (99, 99+11, s)-Nets in Base 8
(99, 99+11, 3617595)-Net over F8 — Constructive and digital
Digital (99, 110, 3617595)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (23, 28, 262155)-net over F8, using
- net defined by OOA [i] based on linear OOA(828, 262155, F8, 6, 5) (dual of [(262155, 6), 1572902, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(828, 262156, F8, 2, 5) (dual of [(262156, 2), 524284, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(82, 9, F8, 2, 2) (dual of [(9, 2), 16, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;16,8) [i]
- linear OOA(826, 262147, F8, 2, 5) (dual of [(262147, 2), 524268, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(826, 524294, F8, 5) (dual of [524294, 524268, 6]-code), using
- trace code [i] based on linear OA(6413, 262147, F64, 5) (dual of [262147, 262134, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(6413, 262144, F64, 5) (dual of [262144, 262131, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(6413, 262147, F64, 5) (dual of [262147, 262134, 6]-code), using
- OOA 2-folding [i] based on linear OA(826, 524294, F8, 5) (dual of [524294, 524268, 6]-code), using
- linear OOA(82, 9, F8, 2, 2) (dual of [(9, 2), 16, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(828, 262156, F8, 2, 5) (dual of [(262156, 2), 524284, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(828, 262155, F8, 6, 5) (dual of [(262155, 6), 1572902, 6]-NRT-code), 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
- digital (23, 28, 262155)-net over F8, using
(99, 99+11, 3879730)-Net in Base 8 — Constructive
(99, 110, 3879730)-net in base 8, using
- (u, u+v)-construction [i] based on
- (23, 28, 524290)-net in base 8, using
- base change [i] based on digital (16, 21, 524290)-net over F16, using
- net defined by OOA [i] based on linear OOA(1621, 524290, F16, 5, 5) (dual of [(524290, 5), 2621429, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1621, 1048581, F16, 5) (dual of [1048581, 1048560, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(1621, 1048576, F16, 5) (dual of [1048576, 1048555, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 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(1621, 1048581, F16, 5) (dual of [1048581, 1048560, 6]-code), using
- net defined by OOA [i] based on linear OOA(1621, 524290, F16, 5, 5) (dual of [(524290, 5), 2621429, 6]-NRT-code), using
- base change [i] based on digital (16, 21, 524290)-net over F16, 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
- (23, 28, 524290)-net in base 8, using
(99, 99+11, large)-Net over F8 — Digital
Digital (99, 110, large)-net over F8, using
- t-expansion [i] based on digital (96, 110, large)-net over F8, using
- 2 times m-reduction [i] based on digital (96, 112, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8112, large, F8, 16) (dual of [large, large−112, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8112, large, F8, 16) (dual of [large, large−112, 17]-code), using
- 2 times m-reduction [i] based on digital (96, 112, large)-net over F8, using
(99, 99+11, large)-Net in Base 8 — Upper bound on s
There is no (99, 110, large)-net in base 8, because
- 9 times m-reduction [i] would yield (99, 101, large)-net in base 8, but