Best Known (53, 53+46, s)-Nets in Base 32
(53, 53+46, 240)-Net over F32 — Constructive and digital
Digital (53, 99, 240)-net over F32, using
- t-expansion [i] based on digital (51, 99, 240)-net over F32, using
- 10 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
- 10 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(53, 53+46, 513)-Net in Base 32 — Constructive
(53, 99, 513)-net in base 32, using
- t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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
- 9 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(53, 53+46, 1194)-Net over F32 — Digital
Digital (53, 99, 1194)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3299, 1194, F32, 46) (dual of [1194, 1095, 47]-code), using
- 156 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 4 times 0, 1, 12 times 0, 1, 23 times 0, 1, 42 times 0, 1, 66 times 0) [i] based on linear OA(3288, 1027, F32, 46) (dual of [1027, 939, 47]-code), using
- construction XX applied to C1 = C([1022,43]), C2 = C([0,44]), C3 = C1 + C2 = C([0,43]), and C∩ = C1 ∩ C2 = C([1022,44]) [i] based on
- linear OA(3286, 1023, F32, 45) (dual of [1023, 937, 46]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,43}, and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3286, 1023, F32, 45) (dual of [1023, 937, 46]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,44], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3288, 1023, F32, 46) (dual of [1023, 935, 47]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,44}, and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3284, 1023, F32, 44) (dual of [1023, 939, 45]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,43], and designed minimum distance d ≥ |I|+1 = 45 [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([1022,43]), C2 = C([0,44]), C3 = C1 + C2 = C([0,43]), and C∩ = C1 ∩ C2 = C([1022,44]) [i] based on
- 156 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 4 times 0, 1, 12 times 0, 1, 23 times 0, 1, 42 times 0, 1, 66 times 0) [i] based on linear OA(3288, 1027, F32, 46) (dual of [1027, 939, 47]-code), using
(53, 53+46, 915752)-Net in Base 32 — Upper bound on s
There is no (53, 99, 915753)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 102295 059240 393517 957764 708251 633543 425983 498880 942312 528371 336090 309794 672292 133948 516417 434746 017947 348031 602359 123333 830531 918525 129215 518174 525648 > 3299 [i]