Best Known (54, 54+23, s)-Nets in Base 32
(54, 54+23, 2982)-Net over F32 — Constructive and digital
Digital (54, 77, 2982)-net over F32, using
- 321 times duplication [i] based on digital (53, 76, 2982)-net over F32, using
- net defined by OOA [i] based on linear OOA(3276, 2982, F32, 23, 23) (dual of [(2982, 23), 68510, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3276, 32803, F32, 23) (dual of [32803, 32727, 24]-code), using
- construction XX applied to C([0,11]) ⊂ C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3267, 32769, F32, 23) (dual of [32769, 32702, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(3243, 32769, F32, 15) (dual of [32769, 32726, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(327, 32, F32, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
- Reed–Solomon code RS(25,32) [i]
- linear OA(321, 2, F32, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to C([0,11]) ⊂ C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(3276, 32803, F32, 23) (dual of [32803, 32727, 24]-code), using
- net defined by OOA [i] based on linear OOA(3276, 2982, F32, 23, 23) (dual of [(2982, 23), 68510, 24]-NRT-code), using
(54, 54+23, 5958)-Net in Base 32 — Constructive
(54, 77, 5958)-net in base 32, using
- 324 times duplication [i] based on (50, 73, 5958)-net in base 32, using
- net defined by OOA [i] based on OOA(3273, 5958, S32, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3273, 65539, S32, 23), using
- 1 times code embedding in larger space [i] based on OA(3272, 65538, S32, 23), using
- discarding parts of the base [i] based on linear OA(25645, 65538, F256, 23) (dual of [65538, 65493, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(25645, 65538, F256, 23) (dual of [65538, 65493, 24]-code), using
- 1 times code embedding in larger space [i] based on OA(3272, 65538, S32, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3273, 65539, S32, 23), using
- net defined by OOA [i] based on OOA(3273, 5958, S32, 23, 23), using
(54, 54+23, 54151)-Net over F32 — Digital
Digital (54, 77, 54151)-net over F32, using
(54, 54+23, large)-Net in Base 32 — Upper bound on s
There is no (54, 77, large)-net in base 32, because
- 21 times m-reduction [i] would yield (54, 56, large)-net in base 32, but