Best Known (53−9, 53, s)-Nets in Base 32
(53−9, 53, 2113582)-Net over F32 — Constructive and digital
Digital (44, 53, 2113582)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 16432)-net over F32, using
- net defined by OOA [i] based on linear OOA(3212, 16432, F32, 4, 4) (dual of [(16432, 4), 65716, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(3212, 32864, F32, 4) (dual of [32864, 32852, 5]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 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
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code) (see above)
- linear OA(321, 1027, F32, 1) (dual of [1027, 1026, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(321, 1027, F32, 1) (dual of [1027, 1026, 2]-code) (see above)
- linear OA(323, 1027, F32, 2) (dual of [1027, 1024, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- Hamming code H(3,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- linear OA(327, 1027, F32, 4) (dual of [1027, 1020, 5]-code), using
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(327, 1023, F32, 4) (dual of [1023, 1016, 5]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 1023, F32, 2) (dual of [1023, 1020, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 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 (see above)
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- linear OA(320, 1027, F32, 0) (dual of [1027, 1027, 1]-code), using
- generalized (u, u+v)-construction [i] based on
- OA 2-folding and stacking [i] based on linear OA(3212, 32864, F32, 4) (dual of [32864, 32852, 5]-code), using
- net defined by OOA [i] based on linear OOA(3212, 16432, F32, 4, 4) (dual of [(16432, 4), 65716, 5]-NRT-code), using
- digital (32, 41, 2097150)-net over F32, using
- net defined by OOA [i] based on linear OOA(3241, 2097150, F32, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3241, 8388601, F32, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, large, F32, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3241, large, F32, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3241, 8388601, F32, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(3241, 2097150, F32, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- digital (8, 12, 16432)-net over F32, using
(53−9, 53, 2621440)-Net in Base 32 — Constructive
(44, 53, 2621440)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (9, 13, 524290)-net over F32, using
- net defined by OOA [i] based on linear OOA(3213, 524290, F32, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(3213, 1048580, F32, 4) (dual of [1048580, 1048567, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(329, 1048576, F32, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(3213, 1048580, F32, 4) (dual of [1048580, 1048567, 5]-code), using
- net defined by OOA [i] based on linear OOA(3213, 524290, F32, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
- (31, 40, 2097150)-net in base 32, using
- base change [i] based on digital (16, 25, 2097150)-net over F256, using
- net defined by OOA [i] based on linear OOA(25625, 2097150, F256, 9, 9) (dual of [(2097150, 9), 18874325, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(25625, 8388601, F256, 9) (dual of [8388601, 8388576, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(25625, 8388601, F256, 9) (dual of [8388601, 8388576, 10]-code), using
- net defined by OOA [i] based on linear OOA(25625, 2097150, F256, 9, 9) (dual of [(2097150, 9), 18874325, 10]-NRT-code), using
- base change [i] based on digital (16, 25, 2097150)-net over F256, using
- digital (9, 13, 524290)-net over F32, using
(53−9, 53, large)-Net over F32 — Digital
Digital (44, 53, large)-net over F32, using
- 3 times m-reduction [i] based on digital (44, 56, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
(53−9, 53, large)-Net in Base 32 — Upper bound on s
There is no (44, 53, large)-net in base 32, because
- 7 times m-reduction [i] would yield (44, 46, large)-net in base 32, but