## DIAMANT problem ## Balanced infinite designs

Construct an infinite random sequence with the following properties:

• 1. The sequence is made of K symbols; e.g., for K=3 and with symbols 1, 2 and 3, the sequence could be:

...2321112133122211232231333311212213231122313211212232233133132...

• 2. Consider subsequences of lengths L; these are denoted SL. E.g., the first S5 explicitly given in the above example is 23211 and the last is 33132. For any subsequence SL of moderate length, the subsubsequences that can be extracted satisfy the following requirements:
• a. All SubS1 have approximately equal frequencies.
• b. All SubS2 have approximately equal frequencies.
• c. SL is non-significantly correlated to time t, where t is the position index in the sequence.

2a. and 2b. are imposed to have relatively equidistributed sequences of symbols; 2c. will further be explained below.

### Context of the problem

The Technische Universiteit Eindhoven, Eurandom and Philips Research (all three in Eindhoven, the Netherlands) are collaborating within a joint-venture on Battery Management. This research is both theoretical and experimental.

Imagine that you charge a battery by a sequence of pulses that are periodically applied, namely at times ..., t-1, t, t+1, ... and that you have K sorts of pulses; furthermore, during the experiment, you record a few relevant parameters describing the battery state.

The experiment must permit us to evaluate the effect of each pulse sort (see 2a), the effect of each pulse sort transition (see 2b) and battery state drifts. These drifts have large time constants, T >> 1. To properly estimate the drifts, you need that SL be little correlated to time t; this leads to above 2c. Other sorts of correlations can clearly be considered.

### Questions

1. Did you ever see such a problem before? Do you know of any field where this problem would occur? (people names, references, ...).
2. How to construct such sequences?
3. What are the interesting properties of such a sequence?

### Example

Using computer brute force, we constructed a sequence that is presently used for the experiments. It is a variation on the above presented theme.

• The infinite sequence consists of the next sequence of 24 symbols, infinitely repeated:

• This sequence has K=4 symbols.
• No symbol is repeated.

### Correspondent

William Rey (w_rey@tvcablenet.be)

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