Best Known (30, 30+21, s)-Nets in Base 32
(30, 30+21, 240)-Net over F32 — Constructive and digital
Digital (30, 51, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 19, 205)-net over F32, using
- net defined by OOA [i] based on linear OOA(3219, 205, F32, 10, 10) (dual of [(205, 10), 2031, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3219, 1025, F32, 10) (dual of [1025, 1006, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 1026, F32, 10) (dual of [1026, 1007, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(3219, 1024, F32, 10) (dual of [1024, 1005, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3217, 1024, F32, 9) (dual of [1024, 1007, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(3219, 1026, F32, 10) (dual of [1026, 1007, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3219, 1025, F32, 10) (dual of [1025, 1006, 11]-code), using
- net defined by OOA [i] based on linear OOA(3219, 205, F32, 10, 10) (dual of [(205, 10), 2031, 11]-NRT-code), using
- 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 (9, 19, 205)-net over F32, using
(30, 30+21, 514)-Net in Base 32 — Constructive
(30, 51, 514)-net in base 32, using
- (u, u+v)-construction [i] based on
- (6, 16, 257)-net in base 32, using
- base change [i] based on digital (0, 10, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 10, 257)-net over F256, using
- (14, 35, 257)-net in base 32, using
- 1 times m-reduction [i] based on (14, 36, 257)-net in base 32, using
- base change [i] based on (8, 30, 257)-net in base 64, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 24, 257)-net over F256, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on (8, 30, 257)-net in base 64, using
- 1 times m-reduction [i] based on (14, 36, 257)-net in base 32, using
- (6, 16, 257)-net in base 32, using
(30, 30+21, 1856)-Net over F32 — Digital
Digital (30, 51, 1856)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3251, 1856, F32, 21) (dual of [1856, 1805, 22]-code), using
- 819 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 11 times 0, 1, 33 times 0, 1, 83 times 0, 1, 161 times 0, 1, 232 times 0, 1, 290 times 0) [i] based on linear OA(3241, 1027, F32, 21) (dual of [1027, 986, 22]-code), using
- construction XX applied to C1 = C([1022,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([1022,19]) [i] based on
- linear OA(3239, 1023, F32, 20) (dual of [1023, 984, 21]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3239, 1023, F32, 20) (dual of [1023, 984, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(3241, 1023, F32, 21) (dual of [1023, 982, 22]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3237, 1023, F32, 19) (dual of [1023, 986, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([1022,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([1022,19]) [i] based on
- 819 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 11 times 0, 1, 33 times 0, 1, 83 times 0, 1, 161 times 0, 1, 232 times 0, 1, 290 times 0) [i] based on linear OA(3241, 1027, F32, 21) (dual of [1027, 986, 22]-code), using
(30, 30+21, 4901895)-Net in Base 32 — Upper bound on s
There is no (30, 51, 4901896)-net in base 32, because
- 1 times m-reduction [i] would yield (30, 50, 4901896)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1809 252190 631104 217592 541005 282658 530335 005245 461625 667800 455329 501614 321536 > 3250 [i]