Best Known (51, 94, s)-Nets in Base 32
(51, 94, 240)-Net over F32 — Constructive and digital
Digital (51, 94, 240)-net over F32, using
- 15 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 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, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(51, 94, 513)-Net in Base 32 — Constructive
(51, 94, 513)-net in base 32, using
- t-expansion [i] based on (46, 94, 513)-net in base 32, using
- 14 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
- 14 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(51, 94, 1268)-Net over F32 — Digital
Digital (51, 94, 1268)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3294, 1268, F32, 43) (dual of [1268, 1174, 44]-code), using
- 229 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 0, 0, 0, 1, 10 times 0, 1, 20 times 0, 1, 38 times 0, 1, 62 times 0, 1, 87 times 0) [i] based on linear OA(3282, 1027, F32, 43) (dual of [1027, 945, 44]-code), using
- construction XX applied to C1 = C([1022,40]), C2 = C([0,41]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([1022,41]) [i] based on
- 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(3280, 1023, F32, 42) (dual of [1023, 943, 43]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,41], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3282, 1023, F32, 43) (dual of [1023, 941, 44]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,41}, and designed minimum distance d ≥ |I|+1 = 44 [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(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([1022,40]), C2 = C([0,41]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([1022,41]) [i] based on
- 229 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 0, 0, 0, 1, 10 times 0, 1, 20 times 0, 1, 38 times 0, 1, 62 times 0, 1, 87 times 0) [i] based on linear OA(3282, 1027, F32, 43) (dual of [1027, 945, 44]-code), using
(51, 94, 1296556)-Net in Base 32 — Upper bound on s
There is no (51, 94, 1296557)-net in base 32, because
- 1 times m-reduction [i] would yield (51, 93, 1296557)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 95 268793 222467 246027 631362 465011 830717 139866 381134 763190 160028 101125 853305 244697 931807 923782 156121 862293 821417 864686 534341 996479 666866 162864 > 3293 [i]