Best Known (48−13, 48, s)-Nets in Base 32
(48−13, 48, 5803)-Net over F32 — Constructive and digital
Digital (35, 48, 5803)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 342)-net over F32, using
- net defined by OOA [i] based on linear OOA(3211, 342, F32, 6, 6) (dual of [(342, 6), 2041, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3211, 1026, F32, 6) (dual of [1026, 1015, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(3211, 1024, F32, 6) (dual of [1024, 1013, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(329, 1024, F32, 5) (dual of [1024, 1015, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(3211, 1026, F32, 6) (dual of [1026, 1015, 7]-code), using
- net defined by OOA [i] based on linear OOA(3211, 342, F32, 6, 6) (dual of [(342, 6), 2041, 7]-NRT-code), using
- digital (24, 37, 5461)-net over F32, using
- net defined by OOA [i] based on linear OOA(3237, 5461, F32, 13, 13) (dual of [(5461, 13), 70956, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3237, 32767, F32, 13) (dual of [32767, 32730, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3237, 32767, F32, 13) (dual of [32767, 32730, 14]-code), using
- net defined by OOA [i] based on linear OOA(3237, 5461, F32, 13, 13) (dual of [(5461, 13), 70956, 14]-NRT-code), using
- digital (5, 11, 342)-net over F32, using
(48−13, 48, 43693)-Net in Base 32 — Constructive
(35, 48, 43693)-net in base 32, using
- base change [i] based on digital (27, 40, 43693)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 43693, F64, 13, 13) (dual of [(43693, 13), 567969, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6440, 262159, F64, 13) (dual of [262159, 262119, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6440, 262159, F64, 13) (dual of [262159, 262119, 14]-code), using
- net defined by OOA [i] based on linear OOA(6440, 43693, F64, 13, 13) (dual of [(43693, 13), 567969, 14]-NRT-code), using
(48−13, 48, 178902)-Net over F32 — Digital
Digital (35, 48, 178902)-net over F32, using
(48−13, 48, 197421)-Net in Base 32
(35, 48, 197421)-net in base 32, using
- base change [i] based on digital (27, 40, 197421)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6440, 197421, F64, 13) (dual of [197421, 197381, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6440, 197421, F64, 13) (dual of [197421, 197381, 14]-code), using
(48−13, 48, large)-Net in Base 32 — Upper bound on s
There is no (35, 48, large)-net in base 32, because
- 11 times m-reduction [i] would yield (35, 37, large)-net in base 32, but