Sphere packings of dimension 1

Introduction

Dimension 1 is the only dimension where the densest packings are known analytically for any size. One-dimensional (1d) sphere packings are simply points on a line. The densest packings are uniformly spaced. The lowest second moment is obtained for zero-mean packings. Hence, the densest packing of any size M can be described by the integers −M+1, −M+3, ..., M−1, possibly rescaled. This modulation format, described in [Gilbert52, p. 517], is often used as a benchmark against which other modulation formats with the same spectral efficiency is compared, which is how the gain parameter G is defined.

In this database, whose primary application is modulation theory, the regular zero-mean packing is called pulse amplitude modulation (PAM). If the packing is shifted so that the lowest level is at zero, it is called amplitude shift keying (ASK). The terminology for these two standard types of 1d packings varies between literature sources.

The highest possible CFM is 2 (i.e., 3.01 dB) and the highest power efficiency is γ = 0. These are both attained by binary PAM, more commonly known as binary phase-shift keying (BPSK). The highest possible gain is G = 0, which by definition is attained by any PAM packing. A 1d packing can never have a positive γ or G.

Database

FileNMdEEnLatβEbCFM [dB]γ [dB]G [dB]Comment
BPSK1_212210.25Y213.010300[Oliver48]. The most common modulation format for coherent communications.
OOK1_21210.50.5Y20.50-3.0103-3.0103On–off keying (OOK) or 2-ASK. The simplest and earliest form of digital modulation, tracing its roots to fire beacons, smoke signals, and telegraphy.
PAM1_31310.6666670.666667Y3.169930.42062-1.24939-2.25950Used in the Fast Ethernet standard 100BASE T-4.
ASK1_31311.666671.66667Y3.169931.05155-5.22879-6.2389-3.9794[Oliver48]
PAM1_414251.25Y42.5-3.9794-3.97940
ASK1_41413.53.5Y41.75-8.45098-8.45098-4.47158[Oliver48], [Sage66]
PAM1_515122Y4.643860.861353-6.0206-5.372410Used in the Fast Ethernet standard 100BASE T-2 and the Gigabit Ethernet standard 1000BASE-T.
ASK1_515166Y4.643862.58406-10.7918-10.1436-4.77121
PAM1_616211.66672.91667Y5.169934.51328-7.65917-6.544930
ASK1_61619.166679.16667Y5.169933.54615-12.6324-11.5182-4.97325
PAM1_717144Y5.614711.42483-9.0309-7.558230
ASK1_71711313Y5.614714.63069-14.1497-12.6771-5.11883
PAM1_8182215.25Y67-10.2119-8.450980
ASK1_818117.517.5Y65.83333-15.4407-13.6798-5.22879
ASK1_10110128.528.5Y6.643868.57935-17.5587-15.3551-5.38391[Oliver48]
PAM1_1611628521.25Y821.25-16.2839-13.27360
PAM1_32132234185.25Y1068.2-22.3172-18.33780
PAM1_6416421365341.25Y12227.5-28.341-23.56980
PAM1_1281128254611365.25Y14780.143-34.3624-28.92170
PAM1_25612562218455461.25Y162730.63-40.3832-34.36260