Best Known (73, 73+23, s)-Nets in Base 4
(73, 73+23, 1032)-Net over F4 — Constructive and digital
Digital (73, 96, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 24, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
(73, 73+23, 1293)-Net over F4 — Digital
Digital (73, 96, 1293)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(496, 1293, F4, 23) (dual of [1293, 1197, 24]-code), using
- 250 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 31 times 0, 1, 46 times 0, 1, 60 times 0, 1, 72 times 0) [i] based on linear OA(486, 1033, F4, 23) (dual of [1033, 947, 24]-code), using
- construction XX applied to C1 = C([1022,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1022,21]) [i] based on
- linear OA(481, 1023, F4, 22) (dual of [1023, 942, 23]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(481, 1023, F4, 22) (dual of [1023, 942, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(486, 1023, F4, 23) (dual of [1023, 937, 24]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(476, 1023, F4, 21) (dual of [1023, 947, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([1022,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1022,21]) [i] based on
- 250 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 31 times 0, 1, 46 times 0, 1, 60 times 0, 1, 72 times 0) [i] based on linear OA(486, 1033, F4, 23) (dual of [1033, 947, 24]-code), using
(73, 73+23, 259111)-Net in Base 4 — Upper bound on s
There is no (73, 96, 259112)-net in base 4, because
- 1 times m-reduction [i] would yield (73, 95, 259112)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1569 280787 342895 200590 572467 441638 004338 180674 435080 854036 > 495 [i]