10 research outputs found
Hadamard partitioned difference families and their descendants
If is a Hadamard difference set (HDS) in , then
is clearly a partitioned
difference family (PDF). Any -PDF will be said of Hadamard-type
if as the one above. We present a doubling construction which,
starting from any such PDF, leads to an infinite class of PDFs. As a special
consequence, we get a PDF in a group of order and three
block-sizes , and , whenever we have a
-HDS and the maximal prime power divisors of are
all greater than
High-rate self-synchronizing codes
Self-synchronization under the presence of additive noise can be achieved by
allocating a certain number of bits of each codeword as markers for
synchronization. Difference systems of sets are combinatorial designs which
specify the positions of synchronization markers in codewords in such a way
that the resulting error-tolerant self-synchronizing codes may be realized as
cosets of linear codes. Ideally, difference systems of sets should sacrifice as
few bits as possible for a given code length, alphabet size, and
error-tolerance capability. However, it seems difficult to attain optimality
with respect to known bounds when the noise level is relatively low. In fact,
the majority of known optimal difference systems of sets are for exceptionally
noisy channels, requiring a substantial amount of bits for synchronization. To
address this problem, we present constructions for difference systems of sets
that allow for higher information rates while sacrificing optimality to only a
small extent. Our constructions utilize optimal difference systems of sets as
ingredients and, when applied carefully, generate asymptotically optimal ones
with higher information rates. We also give direct constructions for optimal
difference systems of sets with high information rates and error-tolerance that
generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication
in the IEEE Transactions on Information Theory. Material presented in part at
the International Symposium on Information Theory and its Applications,
Honolulu, HI USA, October 201
Frame difference families and resolvable balanced incomplete block designs
Frame difference families, which can be obtained via a careful use of
cyclotomic conditions attached to strong difference families, play an important
role in direct constructions for resolvable balanced incomplete block designs.
We establish asymptotic existences for several classes of frame difference
families. As corollaries new infinite families of 1-rotational
-RBIBDs over are
derived, and the existence of -RBIBDs is discussed. We construct
-RBIBDs for , whose
existence were previously in doubt. As applications, we establish asymptotic
existences for an infinite family of optimal constant composition codes and an
infinite family of strictly optimal frequency hopping sequences.Comment: arXiv admin note: text overlap with arXiv:1702.0750
Quadratic Zero-Difference Balanced Functions, APN Functions and Strongly Regular Graphs
Let be a function from to itself and a
positive integer. is called zero-difference -balanced if the
equation has exactly solutions for all non-zero
. As a particular case, all known quadratic planar
functions are zero-difference 1-balanced; and some quadratic APN functions over
are zero-difference 2-balanced. In this paper, we study the
relationship between this notion and differential uniformity; we show that all
quadratic zero-difference -balanced functions are differentially
-uniform and we investigate in particular such functions with the form
, where and where the restriction of to
the set of all non-zero -th powers in is an
injection. We introduce new families of zero-difference -balanced
functions. More interestingly, we show that the image set of such functions is
a regular partial difference set, and hence yields strongly regular graphs;
this generalizes the constructions of strongly regular graphs using planar
functions by Weng et al. Using recently discovered quadratic APN functions on
, we obtain new negative Latin
square type strongly regular graphs
Self-Synchronizing Pulse Position Modulation With Error Tolerance
Pulse position modulation (PPM) is a popular signal modulation technique which converts signals into M-ary data by means of the position of a pulse within a time interval. While PPM and its variations have great advantages in many contexts, this type of modulation is vulnerable to loss of synchronization, potentially causing a severe error floor or throughput penalty even when little or no noise is assumed. Another disadvantage is that this type of modulation typically offers no error correction mechanism on its own, making them sensitive to intersymbol interference and environmental noise. In this paper, we propose a coding theoretic variation of PPM that allows for significantly more efficient symbol and frame synchronization as well as strong error correction. The proposed scheme can be divided into a synchronization layer and a modulation layer. This makes our technique compatible with major existing techniques such as standard PPM, multipulse PPM, and expurgated PPM as well in that the scheme can be realized by adding a simple synchronization layer to one of these standard techniques. We also develop a generalization of expurgated PPM suited for the modulation layer of the proposed self-synchronizing modulation scheme. This generalized PPM can also be used as stand-alone error-correcting PPM with a larger number of available symbols