Best Known (24, 24+18, s)-Nets in Base 32
(24, 24+18, 196)-Net over F32 — Constructive and digital
Digital (24, 42, 196)-net over F32, using
- 2 times m-reduction [i] based on digital (24, 44, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (7, 27, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 17, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(24, 24+18, 455)-Net in Base 32 — Constructive
(24, 42, 455)-net in base 32, using
- base change [i] based on digital (17, 35, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
(24, 24+18, 1254)-Net over F32 — Digital
Digital (24, 42, 1254)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 1254, F32, 18) (dual of [1254, 1212, 19]-code), using
- 220 step Varšamov–Edel lengthening with (ri) = (3, 1, 5 times 0, 1, 18 times 0, 1, 55 times 0, 1, 137 times 0) [i] based on linear OA(3235, 1027, F32, 18) (dual of [1027, 992, 19]-code), using
- construction XX applied to C1 = C([1022,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([1022,16]) [i] based on
- linear OA(3233, 1023, F32, 17) (dual of [1023, 990, 18]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3233, 1023, F32, 17) (dual of [1023, 990, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3235, 1023, F32, 18) (dual of [1023, 988, 19]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3231, 1023, F32, 16) (dual of [1023, 992, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([1022,16]) [i] based on
- 220 step Varšamov–Edel lengthening with (ri) = (3, 1, 5 times 0, 1, 18 times 0, 1, 55 times 0, 1, 137 times 0) [i] based on linear OA(3235, 1027, F32, 18) (dual of [1027, 992, 19]-code), using
(24, 24+18, 1284)-Net in Base 32
(24, 42, 1284)-net in base 32, using
- base change [i] based on digital (17, 35, 1284)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6435, 1284, F64, 3, 18) (dual of [(1284, 3), 3817, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6435, 1366, F64, 3, 18) (dual of [(1366, 3), 4063, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(6435, 1366, F64, 3, 18) (dual of [(1366, 3), 4063, 19]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6435, 1284, F64, 3, 18) (dual of [(1284, 3), 3817, 19]-NRT-code), using
(24, 24+18, 1413909)-Net in Base 32 — Upper bound on s
There is no (24, 42, 1413910)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1645 507880 684641 113083 198323 031198 566108 662421 224252 417830 320118 > 3242 [i]