28 research outputs found
Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance
Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance
d. We study the minimum asymptotic redundancy \rho(q,n,d)=n-log_q A(q,n,d) as n
grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are
designed that improve on the BCH codes and have the lowest asymptotic
redundancy \rho(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this
work, codes of fixed distance that asymptotically surpass BCH codes and the
Gilbert-Varshamov bound were designed only for distances 4,5 and 6.Comment: Submitted to IEEE Trans. on Info. Theor
Strong Singleton type upper bounds for linear insertion-deletion codes
The insertion-deletion codes was motivated to correct the synchronization
errors. In this paper we prove several Singleton type upper bounds on the
insdel distances of linear insertion-deletion codes, based on the generalized
Hamming weights and the formation of minimum Hamming weight codewords. Our
bound are stronger than some previous known bounds. These upper bounds are
valid for any fixed ordering of coordinate positions. We apply these upper
bounds to some binary cyclic codes and binary Reed-Muller codes with any
coordinate ordering, and some binary Reed-Muller codes and one
algebraic-geometric code with certain special coordinate ordering.Comment: 22 pages, references update
Two-batch liar games on a general bounded channel
We consider an extension of the 2-person R\'enyi-Ulam liar game in which lies
are governed by a channel , a set of allowable lie strings of maximum length
. Carole selects , and Paul makes -ary queries to uniquely
determine . In each of rounds, Paul weakly partitions and asks for such that . Carole responds with some
, and if , then accumulates a lie . Carole's string of
lies for must be in the channel . Paul wins if he determines within
rounds. We further restrict Paul to ask his questions in two off-line
batches. We show that for a range of sizes of the second batch, the maximum
size of the search space for which Paul can guarantee finding the
distinguished element is as ,
where is the number of lie strings in of maximum length . This
generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese,
and Deppe. We extend Paul's strategy to solve also the pathological liar
variant, in a unified manner which gives the existence of asymptotically
perfect two-batch adaptive codes for the channel .Comment: 26 page
On nested code pairs from the Hermitian curve
Nested code pairs play a crucial role in the construction of ramp secret
sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum
codes [Ketkar et al. 2006]. The important parameters are (1) the codimension,
(2) the relative minimum distance of the codes, and (3) the relative minimum
distance of the dual set of codes. Given values for two of them, one aims at
finding a set of nested codes having parameters with these values and with the
remaining parameter being as large as possible. In this work we study nested
codes from the Hermitian curve. For not too small codimension, we present
improved constructions and provide closed formula estimates on their
performance. For small codimension we show how to choose pairs of one-point
algebraic geometric codes in such a way that one of the relative minimum
distances is larger than the corresponding non-relative minimum distance.Comment: 28 page
On the Properties of Error Patterns in the ConstantLee Weight Channel
The problem of scalar multiplication applied to vectors is considered in the Lee metric. Unlike in other metrics, the Lee weight of a vector may be increased or decreased by the product with a nonzero, nontrivial scalar.
This problem is of particular interest for cryptographic applications, like for example Lee metric code-based cryptosystems, since an attacker may use scalar multiplication to reduce the Lee weight of the error vector and thus to reduce the complexity of the corresponding generic decoder.
The scalar multiplication problem is analyzed in the asymptotic regime.
Furthermore, the construction of a vector with constant Lee weight using integer partitions is analyzed and an efficient method for drawing vectors of constant Lee weight uniformly at random from the set of all such vectors is given