Best Known (74, 74+31, s)-Nets in Base 32
(74, 74+31, 2187)-Net over F32 — Constructive and digital
Digital (74, 105, 2187)-net over F32, using
- 324 times duplication [i] based on digital (70, 101, 2187)-net over F32, using
- net defined by OOA [i] based on linear OOA(32101, 2187, F32, 31, 31) (dual of [(2187, 31), 67696, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(32101, 32806, F32, 31) (dual of [32806, 32705, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(32101, 32809, F32, 31) (dual of [32809, 32708, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- linear OA(3291, 32769, F32, 31) (dual of [32769, 32678, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3261, 32769, F32, 21) (dual of [32769, 32708, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3210, 40, F32, 9) (dual of [40, 30, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 43, F32, 9) (dual of [43, 33, 10]-code), using
- algebraic-geometric code AG(F,33P) [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- discarding factors / shortening the dual code based on linear OA(3210, 43, F32, 9) (dual of [43, 33, 10]-code), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(32101, 32809, F32, 31) (dual of [32809, 32708, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(32101, 32806, F32, 31) (dual of [32806, 32705, 32]-code), using
- net defined by OOA [i] based on linear OOA(32101, 2187, F32, 31, 31) (dual of [(2187, 31), 67696, 32]-NRT-code), using
(74, 74+31, 4370)-Net in Base 32 — Constructive
(74, 105, 4370)-net in base 32, using
- net defined by OOA [i] based on OOA(32105, 4370, S32, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(32105, 65551, S32, 31), using
- 1 times code embedding in larger space [i] based on OA(32104, 65550, S32, 31), using
- discarding parts of the base [i] based on linear OA(25665, 65550, F256, 31) (dual of [65550, 65485, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding parts of the base [i] based on linear OA(25665, 65550, F256, 31) (dual of [65550, 65485, 32]-code), using
- 1 times code embedding in larger space [i] based on OA(32104, 65550, S32, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(32105, 65551, S32, 31), using
(74, 74+31, 72035)-Net over F32 — Digital
Digital (74, 105, 72035)-net over F32, using
(74, 74+31, large)-Net in Base 32 — Upper bound on s
There is no (74, 105, large)-net in base 32, because
- 29 times m-reduction [i] would yield (74, 76, large)-net in base 32, but