It is shown that, particularly for terrestrial cellular telephony, the interference-suppression feature of CDMA (code division multiple access) can result in a many-fold increase in capacity over analog and even over competing digital techniques. A single-cell system, such as a hubbed satellite network, is addressed, and the basic expression for capacity is developed. The corresponding expressions for a multiple-cell system are derived. and the distribution on the number of users supportable per cell is determined. It is concluded that properly augmented and power-controlled multiple-cell CDMA promises a quantum increase in current cellular capacity. >

On the capacity of a cellular CDMA system

Introduction A fundamental objective of cryptography is to enable two persons to communicate over an insecure channel (a public channel such as the internet) in such a way that any other person is unable to recover their message (called the plaintext ) from what is sent in its place over the channel (the ciphertext ). The transformation of the plaintext into the ciphertext is called encryption , or enciphering. Encryption-decryption is the most ancient cryptographic activity (ciphers already existed four centuries b.c.), but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power. The encryption algorithm takes as input the plaintext and an encryption key K E , and it outputs the ciphertext. If the encryption key is secret, then we speak of conventional cryptography , of private key cryptography , or of symmetric cryptography . In practice, the principle of conventional cryptography relies on the sharing of a private key between the sender of a message (often called Alice in cryptography) and its receiver (often called Bob). If, on the contrary, the encryption key is public, then we speak of public key cryptography . Public key cryptography appeared in the literature in the late 1970s.

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Boolean Models and Methods in Mathematics, Computer Science, and Engineering: Boolean Functions for Cryptography and Error-Correcting Codes

This chapter contains sections titled: Introduction Overview of Multicarrier CDMA Systems Channel Model Performance of MC-CDMA System Performance of Overlapping Multicarrier DS-CDMA Systems Performance of Multicarrier DS-CDMA-I Systems Performance of AMC DS-CDMA Systems Performance of SFH/MC DS-CDMA Systems Chapter Summary and Conclusion 
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Overview of Multicarrier CDMA

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) of all numbers of the form \( a + \sqrt {{ - 1b}} \) with \( a,\;b \in \mathbb{Z} \) and showed that \( \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] \) is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form \( f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 \) with \( D \in \mathbb{Z} \), \( \sqrt {{D\; \notin \;\mathbb{Z}}} \) can be written in the form \( \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) \). Hence the question about the representation of integers by f(a 1, a 2) with \( {a_1},\;{a_2} \in \mathbb{Z} \) can be reformulated as a question of factorization of algebraic numbers of the form \( {a_1} + \sqrt {{D{a_2}}} \). These numbers form a module in the field \( \mathbb{Q}(\sqrt {D} ) \).

Basic Number Theory

Data storage circuits are connected to the bit lines in a one-to-one correspondence. A write circuit writes the data on a first page into a plurality of first memory cells selected simultaneously by a word line. Thereafter, the write circuit writes the data on a second page into the plurality of first memory cell. Then, the write circuit writes the data on the first and second pages into second memory cells adjoining the first memory cells in the bit line direction.

Semiconductor memory device for storing multivalued data

An apparatus and method for generating an encoding matrix for a low density parity check (LDPC) code having a dual-diagonal matrix as a parity check matrix are disclosed. The apparatus and method construct an information sub-matrix of the encoding matrix with a predetermined number of square matrixes according to a predetermined code rate such that each of the square matrixes has columns and rows with a weight of 1 and has a different offset value, combine the square matrixes with the dual-diagonal matrix, and perform inter-row permutation on the information sub-matrix.

Apparatus and method for encoding a low density parity check code

In this correspondence, the constructions and enumerations of all bent functions represented by a polynomial form of f(x)=/spl Sigma//sub i=1//sup n/2-1/c/sub i/Tr(x/sup 1+2(i)/)+c/sub n/2/Tr/sub 1//sup n/ /sup /2/(x/sup 1+2(n/2)/), c/sub i//spl isin//sub 2/ F/sub 2/ are presented for special cases of n. Using an iterative approach, the construction of bent functions of n variables with degree n/2 is also provided using the constructed quadratic bent functions.

Constructions of quadratic bent functions in polynomial forms

For odd n=2l+1 and an integer /spl rho/ with 1/spl les//spl rho//spl les/l, a new family S/sub o/(/spl rho/) of binary sequences of period 2/sup n/-1 is constructed. For a given /spl rho/, S/sub o/(/spl rho/) has maximum correlation 1+2/sup n+2/spl rho/-1/2/, family size 2/sup n/spl rho//, and maximum linear span n(n+1)/2. Similarly, a new family of S/sub e/(/spl rho/) of binary sequences of period 2/sup n/-1 is also presented for even n=2l and an integer /spl rho/ with 1/spl les//spl rho/<l, where maximum correlation, family size, and maximum linear span are 1+2/sup n/2+/spl rho//,2/sup n/spl rho//, and n(n+1)/2, respectively. The new family S/sub o/(/spl rho/) (or S/sub e/(/spl rho/)) contains Boztas and Kumar's construction (or Udaya's) as a subset if m-sequences are excluded from both constructions. As a good candidate with low correlation and large family size, the family S/sub o/(2) is discussed in detail by analyzing its distribution of correlation values.

A new binary sequence family with low correlation and large size

A forward error correction method for decoding coded bits generated by low density parity check matrixes. The method comprises converting each of the coded bits into a log likelihood ratio (LLR) value, and applying the converted values to variable nodes; delivering messages applied to the variable nodes to check nodes; checking a message having a minimum value among the messages, and determining a sign of the message having the minimum value; receiving messages updated in the check nodes, adding up signs of the received messages and a sign of an initial message, applying a weighting factor of 1 when all signs are identical, and when all signs are not identical, updating a message of a variable node by applying a weighting factor; determining LLR of an initial input value; and hard-deciding values of the variable nodes, performing parity check on the hard decision values, and stopping the decoding when no error occurs.

Forward error correction apparatus and method in a high-speed data transmission system

An (N, K) codebook is a set of N unit-norm code vectors in a K-dimensional vector space. For its applications, it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible, meeting the Welch bound equality strictly or asymptotically. In this paper, an (N, K) codebook is constructed from a K × N partial matrix with K <; N, where each code vector is equivalent to a column of the matrix. To obtain the K × N matrix, K rows are selected from a J × N matrix Φ, associated with a binary sequence of length J and Hamming weight K, where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence. It is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of Φ-transform of the binary sequence. Thus, constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its Φ-transform is as small as possible. From the discovery, new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from Fourier and Hadamard matrices associated with binary sequences.

Nam Yul Yu

Papers

Apparatus and method for encoding a low density parity check code

Constructions of quadratic bent functions in polynomial forms

A new binary sequence family with low correlation and large size

Forward error correction apparatus and method in a high-speed data transmission system

A Construction of Codebooks Associated With Binary Sequences