Best Known (77−40, 77, s)-Nets in Base 32
(77−40, 77, 202)-Net over F32 — Constructive and digital
Digital (37, 77, 202)-net over F32, using
- 2 times m-reduction [i] based on digital (37, 79, 202)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 28, 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, 51, 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, 28, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(77−40, 77, 288)-Net in Base 32 — Constructive
(37, 77, 288)-net in base 32, using
- 21 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 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, 70, 288)-net over F128, using
(77−40, 77, 513)-Net over F32 — Digital
Digital (37, 77, 513)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3277, 513, F32, 2, 40) (dual of [(513, 2), 949, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3277, 515, F32, 2, 40) (dual of [(515, 2), 953, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3277, 1030, F32, 40) (dual of [1030, 953, 41]-code), using
- construction XX applied to C1 = C([1021,36]), C2 = C([0,37]), C3 = C1 + C2 = C([0,36]), and C∩ = C1 ∩ C2 = C([1021,37]) [i] based on
- linear OA(3274, 1023, F32, 39) (dual of [1023, 949, 40]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,36}, and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3272, 1023, F32, 38) (dual of [1023, 951, 39]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,37], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3276, 1023, F32, 40) (dual of [1023, 947, 41]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−2,−1,…,37}, and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,36], and designed minimum distance d ≥ |I|+1 = 38 [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([1021,36]), C2 = C([0,37]), C3 = C1 + C2 = C([0,36]), and C∩ = C1 ∩ C2 = C([1021,37]) [i] based on
- OOA 2-folding [i] based on linear OA(3277, 1030, F32, 40) (dual of [1030, 953, 41]-code), using
- discarding factors / shortening the dual code based on linear OOA(3277, 515, F32, 2, 40) (dual of [(515, 2), 953, 41]-NRT-code), using
(77−40, 77, 167011)-Net in Base 32 — Upper bound on s
There is no (37, 77, 167012)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 78 811646 049388 591127 191532 235361 377720 147826 342380 916580 660367 873852 122310 057421 017542 787943 521767 818738 535947 539784 > 3277 [i]