Information Technology Reference
In-Depth Information
The Peak to Sidelobe Level of the Most
Significant Bit of Trace Codes over Galois Rings
Patrick Sole
and Dimitrii Zinoviev
CNRS-I3S,
Les Algorithmes, Euclide B, 2000, route des Lucioles, 06 903 Sophia Antipolis, France
sole@unice.fr
Abstract.
Weighted degree trace codes over even characteristic Galois
rings give binary sequences by projection on their most significant bit
(MSB). Upper bounds on the aperiodic correlation, peak to sidelobe level
(PSL), partial period imbalance and partial period pattern imbalance of
these sequences are derived. The proof technique involves estimates of
incomplete character sums over Galois rings, combining Weil-like bounds
with Fourier transform estimates.
Keywords:
aperiodic autocorrelation, partial period distribution, Galois
rings, MSB, PSL.
1
Introduction
The aperiodic autocorrelation of binary sequences is an important design cri-
terion of binary spreading sequences in a CDMA environment. It is also a
fascinating mathematical invariant in relation with the merit factor [3]. In a
seminal paper [9] Sarwate used Fourier coecient estimates to derive an upper
bound on what would be called later the PSL (Peak to Sidelobe Level) of binary
M
sequences [2]. This bound was extended recently, using similar techniques,
to the Most Significant Bit of M-sequences over rings [10] in the terminology of
Z-D. Dai [1]. Since the proof involves estimates of incomplete character sums it
is very natural to find applications in partial period statistics showing that the
distribution of symbols (or the
r
−
tuples of symbols) in the sequences are close to
uniform. In the present work we generalize the results of [10] to weighted degree
trace codes in primitive length. The case of [10] corresponds in that setting to a
linear polynomial argument of the trace. Our work is also the analogue for the
MSB map of the work [4] for the Gray map.
The material is organized in the following way. Section 2 collects some well-
known definitions on Galois rings. Section 3 contains our version of Sarwate's
DFT bounding technique. Section 4 recalls properties of polynomials over Ga-
lois rings and establishes an important technical lemma (Lemma 4.3). Section 5
studies the uniformity of distribution of symbols zero and one in the above
−
