Best Known (110−17, 110, s)-Nets in Base 32
(110−17, 110, 1310720)-Net over F32 — Constructive and digital
Digital (93, 110, 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 (64, 81, 1048575)-net over F32, using
- net defined by OOA [i] based on linear OOA(3281, 1048575, F32, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3281, 8388601, F32, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3281, large, F32, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3281, large, F32, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3281, 8388601, F32, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(3281, 1048575, F32, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- digital (21, 29, 262145)-net over F32, using
(110−17, 110, 1572864)-Net in Base 32 — Constructive
(93, 110, 1572864)-net in base 32, using
- (u, u+v)-construction [i] based on
- (24, 32, 524289)-net in base 32, using
- net defined by OOA [i] based on OOA(3232, 524289, S32, 8, 8), using
- OA 4-folding and stacking [i] based on OA(3232, 2097156, S32, 8), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on OA(3231, 2097155, S32, 8), using
- OA 4-folding and stacking [i] based on OA(3232, 2097156, S32, 8), using
- net defined by OOA [i] based on OOA(3232, 524289, S32, 8, 8), using
- (61, 78, 1048575)-net in base 32, using
- base change [i] based on digital (48, 65, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- base change [i] based on digital (48, 65, 1048575)-net over F64, using
- (24, 32, 524289)-net in base 32, using
(110−17, 110, large)-Net over F32 — Digital
Digital (93, 110, large)-net over F32, using
- t-expansion [i] based on digital (87, 110, large)-net over F32, using
(110−17, 110, large)-Net in Base 32 — Upper bound on s
There is no (93, 110, large)-net in base 32, because
- 15 times m-reduction [i] would yield (93, 95, large)-net in base 32, but