Best Known (98−15, 98, s)-Nets in Base 32
(98−15, 98, 1547900)-Net over F32 — Constructive and digital
Digital (83, 98, 1547900)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (20, 27, 349529)-net over F32, using
- net defined by OOA [i] based on linear OOA(3227, 349529, F32, 7, 7) (dual of [(349529, 7), 2446676, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3227, 1048588, F32, 7) (dual of [1048588, 1048561, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 1048590, F32, 7) (dual of [1048590, 1048563, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [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(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(322, 14, F32, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 1048590, F32, 7) (dual of [1048590, 1048563, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3227, 1048588, F32, 7) (dual of [1048588, 1048561, 8]-code), using
- net defined by OOA [i] based on linear OOA(3227, 349529, F32, 7, 7) (dual of [(349529, 7), 2446676, 8]-NRT-code), using
- digital (56, 71, 1198371)-net over F32, using
- net defined by OOA [i] based on linear OOA(3271, 1198371, F32, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3271, 8388598, F32, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3271, 8388598, F32, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(3271, 1198371, F32, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- digital (20, 27, 349529)-net over F32, using
(98−15, 98, 1897424)-Net in Base 32 — Constructive
(83, 98, 1897424)-net in base 32, using
- 321 times duplication [i] based on (82, 97, 1897424)-net in base 32, using
- (u, u+v)-construction [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
- (54, 69, 1198371)-net in base 32, using
- net defined by OOA [i] based on OOA(3269, 1198371, S32, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3269, 8388598, S32, 15), using
- discarding factors based on OA(3269, large, S32, 15), using
- discarding parts of the base [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- discarding factors based on OA(3269, large, S32, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3269, 8388598, S32, 15), using
- net defined by OOA [i] based on OOA(3269, 1198371, S32, 15, 15), using
- (21, 28, 699053)-net in base 32, using
- (u, u+v)-construction [i] based on
(98−15, 98, large)-Net over F32 — Digital
Digital (83, 98, large)-net over F32, using
- 7 times m-reduction [i] based on digital (83, 105, large)-net over F32, using
(98−15, 98, large)-Net in Base 32 — Upper bound on s
There is no (83, 98, large)-net in base 32, because
- 13 times m-reduction [i] would yield (83, 85, large)-net in base 32, but