Best Known (43, 78, s)-Nets in Base 32
(43, 78, 240)-Net over F32 — Constructive and digital
Digital (43, 78, 240)-net over F32, using
- 7 times m-reduction [i] based on digital (43, 85, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 32, 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, 53, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 32, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(43, 78, 513)-Net in Base 32 — Constructive
(43, 78, 513)-net in base 32, using
- 12 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
(43, 78, 1266)-Net over F32 — Digital
Digital (43, 78, 1266)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3278, 1266, F32, 35) (dual of [1266, 1188, 36]-code), using
- 228 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 62 times 0, 1, 94 times 0) [i] based on linear OA(3266, 1026, F32, 35) (dual of [1026, 960, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3266, 1024, F32, 35) (dual of [1024, 958, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3264, 1024, F32, 34) (dual of [1024, 960, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 228 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 62 times 0, 1, 94 times 0) [i] based on linear OA(3266, 1026, F32, 35) (dual of [1026, 960, 36]-code), using
(43, 78, 1520635)-Net in Base 32 — Upper bound on s
There is no (43, 78, 1520636)-net in base 32, because
- 1 times m-reduction [i] would yield (43, 77, 1520636)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 78 804558 132894 957076 894228 351993 379002 401645 070530 082765 263328 882417 280644 683950 766818 417514 101407 191255 519382 123297 > 3277 [i]