Best Known (49, 49+42, s)-Nets in Base 32
(49, 49+42, 240)-Net over F32 — Constructive and digital
Digital (49, 91, 240)-net over F32, using
- 12 times m-reduction [i] based on digital (49, 103, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 38, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 65, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 38, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(49, 49+42, 513)-Net in Base 32 — Constructive
(49, 91, 513)-net in base 32, using
- t-expansion [i] based on (46, 91, 513)-net in base 32, using
- 17 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 17 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(49, 49+42, 1176)-Net over F32 — Digital
Digital (49, 91, 1176)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3291, 1176, F32, 42) (dual of [1176, 1085, 43]-code), using
- 138 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 37 times 0, 1, 62 times 0) [i] based on linear OA(3280, 1027, F32, 42) (dual of [1027, 947, 43]-code), using
- construction XX applied to C1 = C([1022,39]), C2 = C([0,40]), C3 = C1 + C2 = C([0,39]), and C∩ = C1 ∩ C2 = C([1022,40]) [i] based on
- linear OA(3278, 1023, F32, 41) (dual of [1023, 945, 42]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,39}, and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(3278, 1023, F32, 41) (dual of [1023, 945, 42]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,40], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(3280, 1023, F32, 42) (dual of [1023, 943, 43]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,40}, and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3276, 1023, F32, 40) (dual of [1023, 947, 41]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,39], and designed minimum distance d ≥ |I|+1 = 41 [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([1022,39]), C2 = C([0,40]), C3 = C1 + C2 = C([0,39]), and C∩ = C1 ∩ C2 = C([1022,40]) [i] based on
- 138 step Varšamov–Edel lengthening with (ri) = (5, 1, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 37 times 0, 1, 62 times 0) [i] based on linear OA(3280, 1027, F32, 42) (dual of [1027, 947, 43]-code), using
(49, 49+42, 932056)-Net in Base 32 — Upper bound on s
There is no (49, 91, 932057)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 93035 363443 343430 338361 940983 753892 506733 561462 873716 119714 144875 048288 774887 323062 356169 853387 552679 882911 035172 950499 162933 958528 163064 > 3291 [i]