Best Known (29−7, 29, s)-Nets in Base 32
(29−7, 29, 350551)-Net over F32 — Constructive and digital
Digital (22, 29, 350551)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 1025)-net over F32, using
- net defined by OOA [i] based on linear OOA(324, 1025, F32, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(324, 1025, F32, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(324, 1025, F32, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- digital (18, 25, 349526)-net over F32, using
- net defined by OOA [i] based on linear OOA(3225, 349526, F32, 7, 7) (dual of [(349526, 7), 2446657, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3225, 1048579, F32, 7) (dual of [1048579, 1048554, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 1048580, F32, 7) (dual of [1048580, 1048555, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(3225, 1048576, F32, 7) (dual of [1048576, 1048551, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(3225, 1048580, F32, 7) (dual of [1048580, 1048555, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3225, 1048579, F32, 7) (dual of [1048579, 1048554, 8]-code), using
- net defined by OOA [i] based on linear OOA(3225, 349526, F32, 7, 7) (dual of [(349526, 7), 2446657, 8]-NRT-code), using
- digital (1, 4, 1025)-net over F32, using
(29−7, 29, 699053)-Net in Base 32 — Constructive
(22, 29, 699053)-net in base 32, using
- 321 times duplication [i] based on (21, 28, 699053)-net in base 32, using
- base change [i] based on digital (13, 20, 699053)-net over F128, using
- net defined by OOA [i] based on linear OOA(12820, 699053, F128, 7, 7) (dual of [(699053, 7), 4893351, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- net defined by OOA [i] based on linear OOA(12820, 699053, F128, 7, 7) (dual of [(699053, 7), 4893351, 8]-NRT-code), using
- base change [i] based on digital (13, 20, 699053)-net over F128, using
(29−7, 29, 1818665)-Net over F32 — Digital
Digital (22, 29, 1818665)-net over F32, using
(29−7, 29, 2086556)-Net in Base 32
(22, 29, 2086556)-net in base 32, using
- 321 times duplication [i] based on (21, 28, 2086556)-net in base 32, using
- base change [i] based on digital (13, 20, 2086556)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12820, 2086556, F128, 7) (dual of [2086556, 2086536, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12820, 2086556, F128, 7) (dual of [2086556, 2086536, 8]-code), using
- base change [i] based on digital (13, 20, 2086556)-net over F128, using
(29−7, 29, large)-Net in Base 32 — Upper bound on s
There is no (22, 29, large)-net in base 32, because
- 5 times m-reduction [i] would yield (22, 24, large)-net in base 32, but