Information and Coding Theory Some applications of error correcting codes. Juris Viksna, 2015.

Information and Information and Coding TheoryCoding Theory

Some applications of error correcting Some applications of error correcting codes.codes.

Juris Viksna, 2015

Data encoding on CDs

[Adapted from H.Hoeve et al]

Data encoding on CDs

[Adapted from D.Salomon]

Data encoding on CDs

CD physical characteristics:

A Compact Disc is made from a 1.2 mm thick disc of almost pure polycarbonate plastic and weighs approximately 16 grammes. A thin layer of Super Purity Aluminium (or rarely gold, used for its data longevity, such as in some limited-edition audiophile CDs) is applied to the surface to make it reflective, and is protected by a film of lacquer.The lacquer is normally printed directly and not with an adhesive label. Common printing methods for compact discs are screen-printing and offset printing.CD data is stored as a series of tiny indentations (pits), encoded in a tightly packed spiral track moulded into the top of the polycarbonate layer. The areas between pits are known as 'lands'. Each pit is approximately 100 nm deep by 500 nm wide, and varies from 850 nm to 3.5 μm in length.

[Adapted from www.wikipedia.org]

Data encoding on CDs

Data storage on CD:

The smallest entity in the CD audio format is called a frame. A frame can accommodate six complete 16-bit stereo samples, i.e. 2×2×6 = 24 bytes. A frame comprises 33 bytes, of which 24 are audio bytes (six full stereo samples), eight CIRC-generated error correction bytes and one subcode byte. The eight bits of a subcode byte are available for control and display. Under Eight-to-Fourteen Modulation (EFM) rules, each data/audio byte is translated into 14-bit EFM words, which alternate with 3-bit merging words. In total we have 33*(14+3) = 561 bits. A 27-bit unique synchronization word is added, so that the number of bits in a frame totals 588.

[Adapted from www.wikipedia.org]

Data encoding on CDs

Data storage on CD:

The synchronization word cannot occur in the normal bit stream, and can thus be used to identify the beginning of a frame. Data on a CD-ROM are organized in both frames and sectors, where a CD-ROM sector contains 98 frames, and holds 98×24 = 2352 (user) bytes, of which 304 bytes are normally used for sector IDs and an additional layer of error correction, leaving 2048 bytes for payload data.The largest entity on a CD is called a track. A CD can contain 99 tracks.

[Adapted from www.wikipedia.org]

Data encoding on CDs

[Adapted from en.kioskea.net]

CD standards

[Adapted from en.kioskea.net]

CD standards

[Adapted from en.kioskea.net]

CD standards

[Adapted from en.kioskea.net]

CD file systems

[Adapted from en.kioskea.net]

Data encoding on CDs

[Adapted from D.Salomon]

Data encoding on CDs

Data encoding on CDs

[Adapted from D.Salomon]

Data encoding on CDs

[Adapted from D.Salomon]

Data encoding on CDs

[Adapted from D.Salomon]

Information coding on CDs

[Adapted from P.Shankar]

Reed-Solomon codes C1 and C2

[Adapted from www.wikipedia.org]

Reed-Solomon codes

Given a finite field F and polynomial ring F[x], let n and k be chosen such that 1 ≤ k ≤ n ≤ | F |. Pick n distinct elements of F, denoted { x1, x2, ... , xn }. Then, the codebook C is created from the tuplets of values obtained by evaluating every polynomial (over F) of degree less than k at each xi; that is:

C is a [n, k, n k+1] code; in other words, it is a linear code of length n (over F) with dimension k and minimum distance nk+1.

C1: n=32, k=28C2: n=28, k=24

Actually these are shortened RS codes obtained starting from [255,251,5] code(s)

GF(28) is probably :) defined by usingpolynomial x8+x4+x3+x2+1

RS codes - a geometric interpretation

[Adapted from P.Shankar]

Interleaving (CIRC)

[Adapted from A.Burr]

Interleaving (CIRC)

[Adapted from D.Salomon]

CIRC

[Adapted from P.Shankar]

Encoding/decoding of CIRC

[Adapted from P.Shankar]

Encoding/decoding of CIRC

[Adapted from H.Hoeve]

CIRC decoding chip. 45 mm2, 12000 gates.

CIRC

[Adapted from P.Shankar]

CIRC

[Adapted from P.Shankar]

CIRC

[Adapted from P.Shankar]

CIRC

[Adapted from P.Shankar]

Additional error correction for CD-ROMs

[Adapted from www.wikipedia.org]

Additional error correction for CD-ROMs

[Adapted from www.wikipedia.org]

ECC - a third layer of error correction codes. Based on RS (26,24) andRS (45,43).