Best Known (97, 132, s)-Nets in Base 4
(97, 132, 531)-Net over F4 — Constructive and digital
Digital (97, 132, 531)-net over F4, using
- 3 times m-reduction [i] based on digital (97, 135, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 45, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 45, 177)-net over F64, using
(97, 132, 1040)-Net over F4 — Digital
Digital (97, 132, 1040)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4132, 1040, F4, 35) (dual of [1040, 908, 36]-code), using
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(4131, 1033, F4, 35) (dual of [1033, 902, 36]-code), using
- construction XX applied to C1 = C([1022,32]), C2 = C([0,33]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([1022,33]) [i] based on
- linear OA(4126, 1023, F4, 34) (dual of [1023, 897, 35]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4126, 1023, F4, 34) (dual of [1023, 897, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4131, 1023, F4, 35) (dual of [1023, 892, 36]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(4121, 1023, F4, 33) (dual of [1023, 902, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [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([1022,32]), C2 = C([0,33]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([1022,33]) [i] based on
- 6 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0) [i] based on linear OA(4131, 1033, F4, 35) (dual of [1033, 902, 36]-code), using
(97, 132, 104272)-Net in Base 4 — Upper bound on s
There is no (97, 132, 104273)-net in base 4, because
- 1 times m-reduction [i] would yield (97, 131, 104273)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7 411090 763774 681542 416058 999324 835269 729255 190238 599033 008743 543088 386533 662708 > 4131 [i]