7 FEHLHABER, L.: 'Statistische Verteilung des Brechwertgradienten in der bodennahen Troposphare'. Deutsche Bundespost report FTZ 455 TBr 43,. 1973.
7 FEHLHABER, L.: 'Statistische Verteilung des Brechwertgradienten in der bodennahen Troposphare'. Deutsche Bundespost report FTZ 455 TBr 43, 1973 8 HALL, M.P.M., and COMER, CM.: 'Statistics of tropospheric radiorefractive-index soundings taken over a 3-year period in the United Kingdom', Proc. IEE, 1969, 116, (5), pp. 685-690 9 AB1LD, B., WENSIEN, H., ARNOLD E., and SCHIKORSKI, W.: 'Uber die Ausbreitung ultrakurzer Wellen jehseits des Horizontes unter besonderer Berilcksichtigung der meteorologischen Einwiikungen', NWDR Techn. Hausmitt., 1952,4, pp. 87-100 10 LEHFELDT, W.: 'Die Ausbreitung der ultrakurzen (quasioptischen) Wellen', AEU, 1949, 3, pp. 339-346 11 GOUGH, M.W.: 'Diurnal influences in tropospheric propagation', Marconi Rev., 1958, 21, pp. 198-212 12 RIDER, G.C.: 'Propagation measurements at 858 Mc/s over paths up to 585 km', ibid., 1958, 21, pp. 184-197 13 GROSSKOPF, J.: 'Uber den augenblicklichen Stand der Forschung auf dem Gebiet der tropospha'rischen Streustrahlung-Pt. 2',NTZ, 1956, 9, pp. 315-329 14 KLINKER, L.: 'Beitrag zum Tagesgang der Feldstarken im Ultrakurzwellenbereich', Z. Metegrol, 1955, 9, pp. 178-191 15 KLINKER, L., and KUHN, U.: 'Uber einige Feldstarkeanstiege im Bereich von 100 und 1100 MHz und ihre meteorologischen Ursachen', Beitr. Phys. Attn., 1961, 34, pp. 15-34 16 KUHN, U.: 'Ein Beitrag zur Uberhorizontausbreitung bei 3500 MHz', Nachrichtentechnik, 1972, 22, pp. 149-151 17 FENGLER.G.: 'Wie beeinflusst das Wetter die Radiowellenausbreitung?'
18 19 20 21 22 23 24 25 26
Beispiele angewandter Forschung 1968/1969, Jahrbuch der FraunhoferGesellschaft, 1969, pp. 53-59 KUHN, U., and OGULEWICZ, S.: 'Propagation measurements at 500MHz over sea for varying meteorological parameters', Proc. IEE, 1970, 117, (5), pp. 879-886 GROSSKOPF, J., and FEHLHABER', L.: 'Zur Deutung der Schwunderscheinungen auf Sichtstrecken im dm- und cm-Wellenbereich'. Deutsche Bundespost technical report 5578, 1965 KUHN, U.: 'Ausbreitungsuntersuchungen an Richtfunkstrecken innerhalb und ausserhalb der optischen Sicht bei 1-1 bis 1-3 GHz', Techn. Mitt. BRF, 1959, 3, pp. 3 2 ^ 1 KUHN, U.: 'Die Ausbreitungsbedingungen an der Richtfunkstrecke CalauKolberg (56 km) bei 3300 MHz', ibid., 1963, 7, pp. 71-77 GUTTEBERG, O., and KJELAAS, A.; "The influence of precipitation and multipath fading on frequencies between 10 and 18 GHz'. AGARD-CP-107, 1973, Paper 13 ABEL, N.: "Some observations of scattering from rain on a 12 GHz transhorizon link', AGARD-CP-107, 1973, Paper 12 GROSSKOPF, J.: 'Wellenausbreitung 1. Hochschultaschenbiicher 141/141a' (Bibliogr. Inst. Mannheim/Wien/Ziirich, 1970) GROSSKOPF, J., and FEHLHABER, L.: 'Feldstarke- und Schwundbeobachtungen auf Landstrecken mit optischer Sicht im dm- urid cm-Wellenbereich', Deutsche Bundespost technical report 5577,1963 FEHLHABER, L., and GROSSKOPF, J.: 'Ausbreitungsmessungen auf einer 69 km langen Sichtstrecke bei 2-5 GHz, 4-2 GHz und 6-7 GHz', Deutsche Bundespost report FTZ A 455 TBr 19, 1970
Correspondence CLASS OF BINARY CODES The table of codes belonging to the class proposed by Hashim and Constantinides [Paper 7187 E, Proc. IEE, 1974, 121, (7), pp. 555—558] invites comparison with codes based on Hadamard matrices, particularly since such matrices constructed by the recursive method can be regarded as the numerical representation of arrays of Walsh functions.A An 8 x 8 Hadamard matrix plus its ones/zeros complement provides 16 code members with mutual distances of at least four. It therefore has the same performance as a code with n = 8, k — 4, d — 4. It is not quite a systematic code (four information digits plus four check digits), as can be seen from the following. All the nonzero rows of the 8 x 8 Hadamard matrix can be generated as linear combinations of the following four rows:
0
1 0
1
0
1
0
0
0
0 0 0 1
0 1
1
0
0 1
0 1
01
1
1
0
0 1
1
1
1
1
0 1
0
0
0
0
0 1
1 1
1 1
0
0
0
1
0
1
1
0
1
0
0
0
1
1
0
1
1
1
(128, 8, 64) (256,9,128) (512,10,256) (1024, 11,512) (2048, 12, 1024)
With regard to the Helgert and Stinaff upper bound, my understanding is that the table which they published0 represents only the performance of the best codes then known, and is not in any way a theoretical upper bound. The absolute minimum value of n — k is given by the Hamming or sphere-packing upper bound, and may be found by enumerating the error patterns which may be corrected and the conditions which may be detected and taking log2 of the sum. Very few codes other than those with d = 3 approach this absolute limit. 28th October 1974
D.A. BELL
Department of Electronic Engineering University of Hull Hull HU6 7RX England
The first three characters of the first three rows constitute an identity matrix, so that the first three digits may be taken as information digits. The fourth row does not contain a one in the fourth place (and the first three rows do not contain zeros in the fourth place), so it cannot be treated as a fourth digit in the usual way. Instead, it means that, if a fourth information digit is present, the second half of the code word representing the sum of the other three digits is complemented. This does not present any serious difficulty in coding. In decoding, the presence of the fourth digit is indicated if the second half of the code word is not identical with the first. Even this limitation is removed if the Hadamard matrix is constructed by Brauer's algorithm,8 which is based on quadratic residues, instead of by the recursive method. The generating matrix is then
0
(129,5,65) (258, 6, 129) (520, 9, 257) (1023,7,513) (2053,9, 1025)
The alternative approach to the construction of binary codes based on Hadamard matrices, as proposed by Prof. Bell, is very interesting indeed. The comparison between the codes derived by this approach and the approach described in our paper is valid and useful. The codes produced from Walsh functions in our paper are indeed of low rate and moreover kin tends to zero for n tending to infinity and, in addition, d/2n tends to 0-25. The alternative approach of Bell improves the rate slightly, as can be seen from the comparison of the codes above; this was expressed in a tentative way in our paper. Perhaps a further useful study could include the asymptotic behaviour of the proposed codes, and futher comparison on these lines may prove to be useful. A.A. HASHIM A.G. CONSTANTINIDES
1 Department of Electrical Engineering Imperial College of Science & Technology Exhibition Road London SW7 2BT England
It is therefore possible to make a direct comparison in terms of length n, number of information digits k and minimum distance d between the new codes and Hadamard codes as follows: Hashim and Constantinides (n, k, d) (8,2,5) (34,4,17) (64, 4, 33) PROC. IEE, Vol 122, No. 1, JANUARY
Hadamard («, k, d) (8,4,4) (32, 6, 16) (64, 7,32) 1975
References
A BELL, D.A.: 'Walsh functions and Hadamard matrixes', Electron. Lett., 1966, 2, pp. 340-341 B BRAUER, A.: 'On a new class of Hadamard determinants', Math. Zeit., 1953,58,pp.219-225 C HELGERT, H.J., and STINAFF, R.D.: 'Minimum-distance bounds for linear binary codes', IEEE. Trans., 1973, IT-19, pp. 344-356
DTC20E 47
