Best Known (71, 71+13, s)-Nets in Base 32
(71, 71+13, 1747630)-Net over F32 — Constructive and digital
Digital (71, 84, 1747630)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (17, 23, 349530)-net over F32, using
- net defined by OOA [i] based on linear OOA(3223, 349530, F32, 6, 6) (dual of [(349530, 6), 2097157, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3223, 1048590, F32, 6) (dual of [1048590, 1048567, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(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(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(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(3223, 1048590, F32, 6) (dual of [1048590, 1048567, 7]-code), using
- net defined by OOA [i] based on linear OOA(3223, 349530, F32, 6, 6) (dual of [(349530, 6), 2097157, 7]-NRT-code), using
- digital (48, 61, 1398100)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (17, 23, 349530)-net over F32, using
(71, 71+13, 2796200)-Net in Base 32 — Constructive
(71, 84, 2796200)-net in base 32, using
- base change [i] based on digital (57, 70, 2796200)-net over F64, using
- net defined by OOA [i] based on linear OOA(6470, 2796200, F64, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(6470, 8388601, F64, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6470, 8388602, F64, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(6421, 4194301, F64, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6421, 8388602, F64, 6) (dual of [8388602, 8388581, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- OOA 2-folding [i] based on linear OA(6421, 8388602, F64, 6) (dual of [8388602, 8388581, 7]-code), using
- linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- linear OOA(6421, 4194301, F64, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(6470, 8388602, F64, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(6470, 8388601, F64, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6470, 2796200, F64, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
(71, 71+13, large)-Net over F32 — Digital
Digital (71, 84, large)-net over F32, using
- t-expansion [i] based on digital (68, 84, large)-net over F32, using
- 2 times m-reduction [i] based on digital (68, 86, large)-net over F32, using
(71, 71+13, large)-Net in Base 32 — Upper bound on s
There is no (71, 84, large)-net in base 32, because
- 11 times m-reduction [i] would yield (71, 73, large)-net in base 32, but