Best Known (105−16, 105, s)-Nets in Base 32
(105−16, 105, 1310720)-Net over F32 — Constructive and digital
Digital (89, 105, 1310720)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (21, 29, 262145)-net over F32, using
- net defined by OOA [i] based on linear OOA(3229, 262145, F32, 8, 8) (dual of [(262145, 8), 2097131, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(3229, 1048580, F32, 8) (dual of [1048580, 1048551, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- OA 4-folding and stacking [i] based on linear OA(3229, 1048580, F32, 8) (dual of [1048580, 1048551, 9]-code), using
- net defined by OOA [i] based on linear OOA(3229, 262145, F32, 8, 8) (dual of [(262145, 8), 2097131, 9]-NRT-code), using
- digital (60, 76, 1048575)-net over F32, using
- net defined by OOA [i] based on linear OOA(3276, 1048575, F32, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3276, 8388600, F32, 16) (dual of [8388600, 8388524, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3276, large, F32, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3276, large, F32, 16) (dual of [large, large−76, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3276, 8388600, F32, 16) (dual of [8388600, 8388524, 17]-code), using
- net defined by OOA [i] based on linear OOA(3276, 1048575, F32, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- digital (21, 29, 262145)-net over F32, using
(105−16, 105, 1572863)-Net in Base 32 — Constructive
(89, 105, 1572863)-net in base 32, using
- (u, u+v)-construction [i] based on
- (23, 31, 524288)-net in base 32, using
- net defined by OOA [i] based on OOA(3231, 524288, S32, 8, 8), using
- OA 4-folding and stacking [i] based on OA(3231, 2097152, S32, 8), using
- discarding factors based on OA(3231, 2097155, S32, 8), using
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(12822, 2097155, F128, 8) (dual of [2097155, 2097133, 9]-code), using
- discarding factors based on OA(3231, 2097155, S32, 8), using
- OA 4-folding and stacking [i] based on OA(3231, 2097152, S32, 8), using
- net defined by OOA [i] based on OOA(3231, 524288, S32, 8, 8), using
- (58, 74, 1048575)-net in base 32, using
- net defined by OOA [i] based on OOA(3274, 1048575, S32, 16, 16), using
- OA 8-folding and stacking [i] based on OA(3274, 8388600, S32, 16), using
- discarding factors based on OA(3274, large, S32, 16), using
- discarding parts of the base [i] based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding parts of the base [i] based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- discarding factors based on OA(3274, large, S32, 16), using
- OA 8-folding and stacking [i] based on OA(3274, 8388600, S32, 16), using
- net defined by OOA [i] based on OOA(3274, 1048575, S32, 16, 16), using
- (23, 31, 524288)-net in base 32, using
(105−16, 105, large)-Net over F32 — Digital
Digital (89, 105, large)-net over F32, using
- t-expansion [i] based on digital (87, 105, large)-net over F32, using
- 5 times m-reduction [i] based on digital (87, 110, large)-net over F32, using
(105−16, 105, large)-Net in Base 32 — Upper bound on s
There is no (89, 105, large)-net in base 32, because
- 14 times m-reduction [i] would yield (89, 91, large)-net in base 32, but