8 research outputs found
Enumerative Coding for Grassmannian Space
The Grassmannian space \Gr is the set of all dimensional subspaces of
the vector space~\smash{\F_q^n}. Recently, codes in the Grassmannian have
found an application in network coding. The main goal of this paper is to
present efficient enumerative encoding and decoding techniques for the
Grassmannian. These coding techniques are based on two different orders for the
Grassmannian induced by different representations of -dimensional subspaces
of \F_q^n. One enumerative coding method is based on a Ferrers diagram
representation and on an order for \Gr based on this representation. The
complexity of this enumerative coding is digit
operations. Another order of the Grassmannian is based on a combination of an
identifying vector and a reduced row echelon form representation of subspaces.
The complexity of the enumerative coding, based on this order, is
digits operations. A combination of the two
methods reduces the complexity on average by a constant factor.Comment: to appear in IEEE Transactions on Information Theor
Asymmetric LOCO Codes: Constrained Codes for Flash Memories
In data storage and data transmission, certain patterns are more likely to be
subject to error when written (transmitted) onto the media. In magnetic
recording systems with binary data and bipolar non-return-to-zero signaling,
patterns that have insufficient separation between consecutive transitions
exacerbate inter-symbol interference. Constrained codes are used to eliminate
such error-prone patterns. A recent example is a new family of
capacity-achieving constrained codes, named lexicographically-ordered
constrained codes (LOCO codes). LOCO codes are symmetric, that is, the set of
forbidden patterns is closed under taking pattern complements. LOCO codes are
suboptimal in terms of rate when used in Flash devices where block erasure is
employed since the complement of an error-prone pattern is not detrimental in
these devices. This paper introduces asymmetric LOCO codes (A-LOCO codes),
which are lexicographically-ordered constrained codes that forbid only those
patterns that are detrimental for Flash performance. A-LOCO codes are also
capacity-achieving, and at finite-lengths, they offer higher rates than the
available state-of-the-art constrained codes designed for the same goal. The
mapping-demapping between the index and the codeword in A-LOCO codes allows
low-complexity encoding and decoding algorithms that are simpler than their
LOCO counterparts.Comment: 9 pages (double column), 0 figures, accepted at the Annual Allerton
Conference on Communication, Control, and Computin
Efficient Constrained Codes That Enable Page Separation in Modern Flash Memories
The pivotal storage density win achieved by solid-state devices over magnetic
devices recently is a result of multiple innovations in physics, architecture,
and signal processing. Constrained coding is used in Flash devices to increase
reliability via mitigating inter-cell interference. Recently,
capacity-achieving constrained codes were introduced to serve that purpose.
While these codes result in minimal redundancy, they result in non-negligible
complexity increase and access speed limitation since pages cannot be read
separately. In this paper, we suggest new constrained coding schemes that have
low-complexity and preserve the desirable high access speed in modern Flash
devices. The idea is to eliminate error-prone patterns by coding data either
only on the left-most page (binary coding) or only on the two left-most pages
(-ary coding) while leaving data on all the remaining pages uncoded. Our
coding schemes are systematic and capacity-approaching. We refer to the
proposed schemes as read-and-run (RR) constrained coding schemes. The -ary
RR coding scheme is introduced to limit the rate loss. We analyze the new RR
coding schemes and discuss their impact on the probability of occurrence of
different charge levels. We also demonstrate the performance improvement
achieved via RR coding on a practical triple-level cell Flash device.Comment: 30 pages (single column), 5 figures, submitted to the IEEE
Transactions on Communications (TCOM). arXiv admin note: substantial text
overlap with arXiv:2111.0741
Constant-Weight and Constant-Charge Binary Run-Length Limited Codes
Constant-weight and constant-charge binary sequences with constrained run
length of zeros are introduced. For these sequences, the weight and the charge
distribution are found. Then, recurrent and direct formulas for calculating the
number of these sequences are obtained. With considering these numbers of
constant-weight and constant-charge RLL sequences as coefficients of convergent
power series, generating functions are derived. The fact, that generating
function for enumerating constant-charge RLL sequences does not have a closed
form, is proved. Implementation of encoding and decoding procedures using
Cover's enumerative scheme is shown. On the base of obtained results, some
examples, such as enumeration of running-digital-sum (RDS) constrained RLL
sequences or peak-shifts control capability are also provided.Comment: 29 pages, submitted to IEEE Transactions on Information Theory. This
paper is a corrected version of a paper with the same title that appeared on
the arXiv in Feb. 2009. The major change is in Section VI, in which
Subsection D is now well define