The average bit erasure probability of a binary linear code ensemble under maximum a-posteriori probability (MAP) decoding over binary erasure channel (BEC) can be calculated with the average support weight distribution of the ensemble via the EXIT function and the shortened information function. In this paper, we formulate the relationship between the average bit erasure probability under MAP ...

The Nordstrom-Robinson code, a nonlinear binary code of length 16 and minimal Hamming distance 6, is the binary image of the octacode, a linear self-dual code over 4 of length 8 and minimal Lee distance 6. Since the octacode is the 4-analogue of a Hamming code, this provides an extremely simple definition of the Nordstrom-Robinson code. A different version of this paper appeared in: Coding and ...

A code ${\cal C}$ is $\Z_2\Z_4$-additive if the set of coordinates can be partitioned into two subsets $X$ and $Y$ such that the punctured code of ${\cal C}$ by deleting the coordinates outside $X$ (respectively, $Y$) is a binary linear code (respectively, a quaternary linear code). In this paper $\Z_2\Z_4$-additive codes are studied. Their corresponding binary images, via the Gray map, are $\Z...

Independently developed codebases typically contain many segments of code that perform same or closely related operations (semantic clones). Finding functionally equivalent segments enables applications like replacing a segment by a more efficient or more secure alternative. Such related segments often have different interfaces, so some glue code (an adapter) is needed to replace one with the o...

Network security is very important when sending confidential data through the network. Cryptography is the science of hiding information, and a combination of cryptography solutions with cognitive science starts a new branch called cognitive cryptography that guarantee the confidentiality and integrity of the data. Brain signals as a biometric indicator can convert to a binary code which can be...

| In the late 50s Slepian introduced a structure theory for linear, binary codes and proved that every such code was uniquely the sum of indecomposable codes. He had hoped to produce a canonical form for the generator matrix of an indecompos-able code so that he might read oo the properties of the code from such a matrix, but such a program proved impossible. We here work over an arbitrary eld ...

We give a new construction of the extended binary Golay code. The construction is carried out by taking the Gray image of a self-dual linear code over the ring R = F2+uF2+vF2+uvF2 of length 6 and size 212. Writing a typical generating matrix of the form [I3|A], with A being a 3× 3 matrix over R, and finding some dependencies among the entries of A, we are able to set a general form for the gene...

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson , Kerdock, Preparata, Goethals, and Delsarte-Goethals . It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z4, the integers mod 4 (although this requires a slight modificat...

Proof. Let C be a binary code with minimum relative distance δ. Suppose x ∈ {0, 1} is arbitrary and c1, c2, . . . , cL ∈ C are codewords such that for all i, ∆ (x, ci) < ρn. Since each ci is a codeword, we have that ∆ (ci, cj) ≥ δn for all i 6= j. Let Φ : {0, 1} → Rn be the map sending 0 to 1 and 1 to −1. Define u = Φ (x), and, for all i, define vi = Φ (ci). Let ū = 1 √ n u denote the normaliza...