Best Known (73−38, 73, s)-Nets in Base 32
(73−38, 73, 202)-Net over F32 — Constructive and digital
Digital (35, 73, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 26, 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 (9, 47, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (7, 26, 98)-net over F32, using
(73−38, 73, 288)-Net in Base 32 — Constructive
(35, 73, 288)-net in base 32, using
- 18 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 65, 288)-net over F128, using
(73−38, 73, 490)-Net over F32 — Digital
Digital (35, 73, 490)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3273, 490, F32, 2, 38) (dual of [(490, 2), 907, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3273, 515, F32, 2, 38) (dual of [(515, 2), 957, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3273, 1030, F32, 38) (dual of [1030, 957, 39]-code), using
- construction XX applied to C1 = C([1021,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([1021,35]) [i] based on
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,34}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3268, 1023, F32, 36) (dual of [1023, 955, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3272, 1023, F32, 38) (dual of [1023, 951, 39]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,35}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3266, 1023, F32, 35) (dual of [1023, 957, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- Reed–Solomon code RS(31,32) [i]
- discarding factors / shortening the dual code based on linear OA(321, 32, F32, 1) (dual of [32, 31, 2]-code), using
- 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 XX applied to C1 = C([1021,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([1021,35]) [i] based on
- OOA 2-folding [i] based on linear OA(3273, 1030, F32, 38) (dual of [1030, 957, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(3273, 515, F32, 2, 38) (dual of [(515, 2), 957, 39]-NRT-code), using
(73−38, 73, 155157)-Net in Base 32 — Upper bound on s
There is no (35, 73, 155158)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 75 161952 887019 877828 323796 306257 034865 719663 098774 945235 295457 670481 091117 162227 961757 293195 794628 966439 239790 > 3273 [i]